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As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Bericht in F. d. M. XXV. 1893/94. 1632 ff., JFM 25.1632.01. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. In archaeology, the reconstruction of the relative chronology of objects (e.g. graves) is often based on absence/presence of information about finds (e.g. grave goods). Traditionally, this task is known as seriation. In this article the task is tackled by formulating a stochastic model for the relationship between the underlying grave order and the observed incidences and by analysing the data using the Bayesian method. In selecting a prior distribution the attempt has been to reflect the archaeological context, especially a potential preselection of specific types of finds suitable for the seriation task. In contrast to established methods for seriation, such as correspondence analysis, it is possible directly to describe the variability of the estimated order by analysing the posterior distribution of the order. Because the order of the graves is a non-numerical and high-dimensional parameter, special techniques for the analysis of the posterior distribution are required. Construction of a Markov chain Monte Carlo method to approximate the posterior distribution is also partially non-standard, since the distribution can be multi-modal and because a huge number of nuisance parameters are introduced to avoid parametric assumptions on the shape of the distribution of the types through time. An example illustrates the techniques and demonstrates the need for a sensitivity analysis in this setting. The framework of our approach can easily be extended either to adjust for known factors which influence the absence/presence or in order to incorporate prior information on the grave order. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The aim of this paper is to give a proof of the existence and uniqueness theorem of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra.
Let \({\mathfrak g}\) be a symmetrizable Kac-Moody Lie algebra and \(U_ q({\mathfrak g})\) be the \(q\)-analogue of the universal enveloping algebra \(U({\mathfrak g})\). For an integrable \(U_ q({\mathfrak g})\)-module \(M\), the endomorphisms \(\tilde e_ i\) and \(\tilde f_ i\) of \(M\) are introduced. Let \(A\) be the subring of \(\mathbb{Q}(q)\) consisting of rational functions regular at \(q=0\). A pair \((L,B)\) is called a (lower) crystal base of \(M\) if it satisfies the conditions: (1) \(L\) is a free sub-A-module of \(M\) such that \(M\simeq\mathbb{Q}(q)\otimes_ AL\); (2) \(B\) is a base of the \(\mathbb{Q}\)-vector space \(L/qL\); (3) \(\tilde e_ iL\subset L\) and \(\tilde f_ iL\subset L\) for any \(i\); (4) \(\tilde e_ iB\subset B\cup\{0\}\) and \(\tilde f_ iB\subset B\cup\{0\}\); (5) \(L=\bigoplus_{\lambda\in P}L_ \lambda\) and \(B=\bigcup_{\lambda\in P}B_ \lambda\), where \(P\) is the weight lattice and \(L_ \lambda=L\cap M_ \lambda\), \(B_ \lambda=B\cap(L_ \lambda/qL_ \lambda)\); (6) For \(b,b'\in B\), \(b'=\tilde f_ ib\) if and only if \(b=\tilde e_ ib'\). For a dominant integral weight \(\lambda\), let \(V(\lambda)\) denote the irreducible \(U_ q({\mathfrak g})\)-module with highest weight \(\lambda\). Let \(u_ \lambda\) be the highest weight vector of \(V(\lambda)\). Let \(L(\lambda)\) be the smallest sub-\(A\)-module of \(V(\lambda)\) that contains \(u_ \lambda\) and that is stable by the actions of \(\tilde f_ i\). Let \(B(\lambda)\) be the subset of \(L(\lambda)/qL(\lambda)\) consisting of the nonzero vectors of the form \(\tilde f_{i_ 1}\dots\tilde f_{i_ k}u_ \lambda\mod qL(\lambda)\). Then \((L(\lambda),B(\lambda))\) is a crystal base of \(V(\lambda)\) (Theorem 2). Let \(M\) be an integrable \(U_ q({\mathfrak g})\)- module such that \(M=\bigoplus_{\lambda\in F-Q_ +}M_ \lambda\) for a finite subset \(F\) of \(P\), where \(Q_ +=\oplus\mathbb{N}\alpha_ i\), and let \((L,B)\) be a crystal base of \(M\). Then there exists an isomorphism \(M\simeq\bigoplus_ j V(\lambda_ j)\) by which \((L,B)\) is isomorphic to \(\bigoplus_ j (L(\lambda_ j),B(\lambda_ j))\) (Theorem 3). Theorem 2 is proved by the induction on height of weights. The good behavior of crystal bases under the tensor product plays a crucial role in the course of the proof.
In the second part of the paper the author constructs a base named global crystal base of any highest weight irreducible integrable \(U_ q({\mathfrak g})\)-module. In the case of \(A_ n\), \(D_ n\) and \(E_ n\), this coincides with the canonical base introduced by \textit{G. Lusztig} [J. Algebra 131, 466-475 (1990; Zbl 0698.16007)]. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. In this paper, we propose a defective model induced by a frailty term for modeling the proportion of cured. Unlike most of the cure rate models, defective models have advantage of modeling the cure rate without adding any extra parameter in model. The introduction of an unobserved heterogeneity among individuals has bring advantages for the estimated model. The influence of unobserved covariates is incorporated using a proportional hazard model. The frailty term assumed to follow a gamma distribution is introduced on the hazard rate to control the unobservable heterogeneity of the patients. We assume that the baseline distribution follows a Gompertz and inverse Gaussian defective distributions. Thus we propose and discuss two defective distributions: the defective gamma-Gompertz and gamma-inverse Gaussian regression models. Simulation studies are performed to verify the asymptotic properties of the maximum likelihood estimator. Lastly, in order to illustrate the proposed model, we present three applications in real data sets, in which one of them we are using for the first time, related to a study about breast cancer in the A.C.Camargo Cancer Center, S茫o Paulo, Brazil. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Feynman diagrams enable us to find the relevant expressions in a perturbative expansion to the equations describing the dynamics of a quantum electrodynamic system. From the very conception of the process as a sequence of particle creations and annihilations, one can draw a scheme relating these creations and annihilation events. The diagram thus obtained can then be transmogrified, element by element, into a complex mathematical expression, the evaluation of which yields an observable quantity characterizing the physical system. The aim of this book is to reconstruct the route that led Feynman, between approximately 1946 and 1948, to devise his new methods of diagrams and to evaluate what was achieved.
The book consists of 7 chapters. The core of the book are Chapters 2--6. In Chapter 2 the author discusses the diagrams used by \textit{H. Euler} [Ann. der Physik, V. F. 26, 398--448 (1936; Zbl 0014.23706)] and by Ziro Koba and Gyo Takeda in the latter half of the 1940s as precurssors of diagrammatic representations. By doing so, he discerns the change in the conception and representation of diagrammatic representations brought out by Feynman and Dyson.
Chapter 3 is concerned with Feynman's PhD thesis. In the early 1940s, when Feynman was a graduate student, one of the most pressing problems in theoretical physics was the fact that infinite quantities arose from some of the principles of electrodynamics in both classical and any attempted quantum theory. Feynman's strategy was first to establish a divergence-free classical electrodynamics and then to quantize it. \textit{R. P. Feynman} and his supervisor \textit{J. A. Wheeler} [Rev. Mod. Phys. 21, 425--433 (1949; Zbl 0034.27801)] had already developed an alternative theory of classical electrodynamics with the desired feature that awaited quantization by the time Feynman had started working on his thesis.
The standard procedure for quantizing a classical theory was to interpret the classical Hamiltonian function as an operator in a Hilbert space of standard vectors. The problem with quantizing the Wheeler-Feynman theory of electrodynamics was that it could not be formulated by specifying a Hamiltonian function. Therefore Feynman needed a method to quantize physical systems whose classical descriptions could not be given by Hamiltonian functions. Feynman based his quantization procedure on Paul Dirac's considerations on how to construct a quantum theory out of the Lagrangian formulation of classical mechanics, though the story was not so direct. To Feynman with some misunderstanding, Dirac appeared rather vague about the exact nature of the relationship between the exponential \(e^{\frac{iL\delta t}{\hbar}}\) of the Lagrangian function \(L\) and the function \((q_{t+\delta t}|q_t)\) that related the state descriptions in terms of the coordinates at two different times in an infinitesimal distance. To have a relationship between the wave functions at two times infinitesimally close did not, however, solve Feynman's original problems, which undoubtedly included calculating the probability of transitions from one state at time \(t_0\) to another state at a later time \(T\) with no restrictions on the difference \(T-t_0\). Feynman constructed wave functions inductively by dividing the interval \([t_0,T] \) into a very large number of infinitesimal intervals. It was expected that, in the limit of infinitesimally small intervals, the appropriate relationship would turn into an equality and the sum in the exponential could be replaced with an appropriate integral. The problem is where integrations over infinitely many variables, infinitely close to each other, should be performed. Neither Feynman nor anybody else so far has been able to define such a procedure precisely and to justify rigorously the sum with an integration. Feynman did not consider himself to be in a position to answer the difficult mathematical questions as to the conditions under which the limiting process of subdividing the time scale actually converges. He was satisfied to give a qualitative description of the limiting process and to point out the similarity between the sum in the exponential and a definition of the corresponding integral. The results of Feynman's thesis were non-relativistic throughout.
Chapter 4 is concerned with Feynman's great struggle to look for a physical system, the appropriate description of which would satisfy the Dirac equation. Feynman recognized that the quivering feature of the electron would fit the extension of his alternative formulation to relativistic systems perfectly by finding that the mainly relevant paths were of a type familiar in the study of Brownian motion. The author shows in Chapters 4 and 5 that the first Feynman diagrams (i.e., before Dyson's intervention) are a product of Feynman's efforts to understand the Dirac equation not just in a mathematical way. Feynman's aim was to describe Dirac's well-known theory in alternative ways. Feynman believed that the equations had to be completed by pictures, and several pictures were possible for the same equations. He tried to interpret the known equations in such a way that it became clear which assumption in the theory was causing the inconsistent conclusions in the troublesome cases. \textit{R. P. Feynman}'s alternative theory appeared in [Phys. Rev., II. Ser. 76, 749--759 (1949; Zbl 0037.12406); Phys. Rev., II. Ser. 76, 769--789 (1949; Zbl 0038.13302)].
Chapter 6 deals with Dyson's elaboration of Feynman diagrams. Feynman gave the first public presentation of his alternative formulation of quantum electrodynamics (QED) at a conference on physics sponsored by the National Academy of Sciences and held at Pocono Manor, Pennsylvania, during March 30 and April 1, 1948. It was a complete disaster, for the audience failed to understand how Feynman could possibly justify his results, even though admitting that the derived results were perfectly correct. In [Phys. Rev., II. Ser. 75, 486--502 (1949; Zbl 0032.23702)], \textit{F. J. Dyson}, being familiar with the more conventional method of second quantization, showed that the three theories of R. P. Feynman, \textit{J. Schwinger} [Phys. Rev., II. Ser. 74, 1439--1461 (1948; Zbl 0032.09404)] and Japanese physicists \textit{S. Tomonaga} [Prog. Theor. Phys. 1, 27--42 (1946; Zbl 0038.13101)] and \textit{Z. Koba, T. Tati} and \textit{S. Tomonaga} [Prog. Theor. Phys. 2, 101--116, 198--208 (1947; Zbl 0038.13102)] to remove QED's uninterpretable divergences were equivalent. In [Phys. Rev., II. Ser. 75, 1736--1755 (1949; Zbl 0033.14201)], \textit{F. J. Dyson} extended his systematization of Feynman's methods to include the treatment of problems with several particles in the initial and final states, instead of only one as in the previous paper. Herein Dyson presented Feynman's theory as an \(S\)-matrix theory in the tradition of \textit{W. Heisenberg} [Z. Phys. 120, 513--538 (1943; Zbl 0028.27901); Z. Phys. 120, 673--702 (1943; Zbl 0028.27902)]. The author remarks that Dyson introduced a new equivalence relation among graphs, which enabled him to present the algorithm for evaluating matrix elements more effectively and allowed for fewer redundancies in the diagrams' articulation of the phenomena.
The book is highly readable for physicists as well as for philosophers of physics and physics-oriented mathematicians. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Population size estimation with discrete or nonparametric mixture models is considered, and reliable ways of construction of the nonparametric mixture model estimator are reviewed and set into perspective. Construction of the maximum likelihood estimator of the mixing distribution is done for any number of components up to the global nonparametric maximum likelihood bound using the EM algorithm. In addition, the estimators of Chao and Zelterman are considered with some generalisations of Zelterman's estimator. All computations are done with CAMCR, a special software developed for population size estimation with mixture models. Several examples and data sets are discussed and the estimators illustrated. Problems using the mixture model-based estimators are highlighted. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Da die neueren Versuche s盲mtlich das Resultat haben, da脽\ auch ein Einflu脽\ zweiter Ordnung der Erdbewegung nicht zu konstatieren ist, hat Verf. es f眉r n枚tig gefunden, der von ihm und \textit{FitzGerald} gemachten Hypothese eine allgemeinere Grundlage zu geben. Er stellt zun盲chst die Grundgleichungen der Elektronentheorie auf f眉r ein mit einer geringeren Geschwindigkeit als der des Lichtes sich bewegendes System und transformiert dann die Gleichungen auf ein System, das gegen das erste in der Bewegungsrichtung deformiert ist. Er erh盲lt hierdurch Gleichungen, die ihm gestatten, die in dem einen Felde gegebenen Punkte, bzw. Funktionen sofort auch in dem anderen Felde zu finden. Hiernach f眉hrt er die Hypothese ein, da脽\ die Elektronen ihre Dimensionen in der Bewegung dieser Deformation entsprechend 盲ndern, w盲hrend sie in der Ruhe Kugeln sind, und da脽\ die Kr盲fte, die zwischen ungeladenen Partikeln und zwischen solchen und Elektronen bestehen, in gleicher Weise wie die elektrischen Kr盲fte in einem elektrostatischen System durch Translation beeinflu脽t werden. Es wird nun das elektromagnetische Moment eines einzelnen Elektrons berechnet; f眉r die \textit{Abraham}sche quasi-station盲re Bewegung ergibt sich dann eine rein elektromagnetische Masse des Elektrons. Dann wird der Einflu脽\ der Bewegung auf optische Ph盲nomene betrachtet, wobei der Verf. zu dem Schlusse kommt, da脽\ in der Deformation \((l,l,kl)\) das \(l=\text{const.}\) sein mu脽, und die Anwendung auf die 眉brigen neueren Versuche f眉hrt zu der allgemeinsten Hypothese, da脽\ ``die Massen aller Partikel durch die Bewegung in gleicher Weise beeinflu脽t werden, wie die elektromagnetischen Massen der Elektronen''. Im weiteren wird die Theorie an \textit{Kaufmanns} Tabellen gepr眉ft und gibt dabei ungef盲hr gleich gute 脺bereinstimmung, wie die \textit{Kaufmann}schen Formeln. Zum Schlu脽\ wird noch der Versuch von \textit{Trouton} diskutiert. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. In this paper we show how the formal framework introduced in PAMR (Process Algebra for the Management of Resources) can be included into a notion of Extended Finite State Machines (in short EFSMs). First, we give the definition of process. Following the lines of PAMR, a process consists not only of information about its behavior but also of information about the preference for resources. This information will be encoded into a model based on EFSMs. In contrast with the original definition of PAMR, a notion of time is included in our processes, that is, transitions take time to be performed. Systems are defined as the composition of several processes. We present different implementation relations, depending on the interpretation of time, and we relate them. Finally, we study how tests cases are defined and applied to implementations. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The author proposes a ``homological mirror conjecture'' relating mirror symmetry to general structures of homological algebra. Let \(V\) be a \(2n\)-dimensional symplectic manifold with \(c_1 (V) = 0\) and \(W\) be a dual \(n\)-dimensional complex algebraic manifold. Let \(LV\) be the space of pairs \((x,L)\), where \(x\) is a point of \(V\) and \(L\) is a Lagrangian subspace of \(T_x V\). There exists a \(\mathbb Z\)-covering \(\widetilde {LV}\) of \(LV\) inducing a universal cover of each fiber. \textit{K. Fukaya} [Morse homotopy, \(A_\infty\)-category and Floer homologies, MSRI preprint No. 020-94 (1993), see also Kim, Hong-Jong (ed.), Proceedings of the GARC workshop on geometry and topology '93 held at the Seoul National University, Seoul, Korea, July 1993. Seoul: Seoul National University, Lect. Notes Ser., Seoul. 18, 1--102 (1993; Zbl 0853.57030)], based on ideas of Donaldson, Floer and Segal, constructed an \(A_\infty\)-category \(F(V)\) having as objects the Lagrangian submanifolds \({\mathcal L} \subset V\) endowed with a continuous lift \({\mathcal L} \to \widetilde {LV}\) of the map \({\mathcal L} \to LV\). (An \(A_\infty\)-category \(C\) is a collection of objects and \(\mathbb Z\)-graded spaces of morphisms \(\Hom_C (X,Y)\) endowed with higher compositions of morphisms satisfying relations similar to the defining relations of \(A_\infty\)-algebras; an \(A_\infty\)-algebra is a concept introduced by \textit{J. D. Stasheff} [Trans. Am. Math. Soc. 108, 275--292, 293--312 (1963; Zbl 0114.39402)].)
The conjecture says that the derived category \(D^b (F(V))\) (or a suitable enlarged one) is equivalent to the derived category \(D^b (\text{Coh} (W))\) of coherent sheaves on \(W\). | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The descent algebra of a finite Coxeter group \(W\) is a subalgebra of the group algebra defined by Solomon. Descent algebras of symmetric groups have properties that are not shared by other Coxeter groups. For instance, the natural map from the descent algebra of a symmetric group to its character ring is a surjection with kernel equal the Jacobson radical. Thus, the descent algebra implicitly encodes information about the representations of the symmetric group, and a complete set of primitive idempotents in the character ring leads to a decomposition of the group algebra into a sum of right ideals indexed by partitions. Stanley asked whether this decomposition of the regular representation of a symmetric group could be realized as a sum of representations induced from linear characters of centralizers. This question was answered positively by Bergeron, Bergeron, and Garsia, using a connection with the free Lie algebra on \(n\) letters, and independently by Douglass, Pfeiffer, and R枚hrle, who connected the decomposition with the configuration space of \(n\)-tuples of distinct complex numbers. The Mantaci-Reutenauer algebra of a hyperoctahedral group is a subalgebra of the group algebra that contains the descent algebra. Bonnaf茅 and Hohlweg showed that the natural map from the Mantaci-Reutenauer algebra to the character ring is a surjection with kernel equal the Jacobson radical. In 2008, Bonnaf茅 asked whether the analog to Stanley's question about the decomposition of the group algebra into a sum of induced linear characters holds. In this paper, we give a positive answer to Bonnaf茅's question by explicitly constructing the required linear characters. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The concept of a topological field theory is extended to encompass structures associated with manifolds of codimension \(> 1\). When all the manifolds involved are considered triangulated, it is seen that such structures may be constructed from a finite quantity of data, most conveniently viewed as associated with polyhedra and their decompositions. The special cases of 2 and 3 dimensions are briefly considered, the relations with structures of higher categories, algebras and vector spaces, becoming clear. A more detailed account is currently in preparation. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Wir berichten 眉ber dieses uns nicht zugegangene Buch nach der Anzeige von \textit{Archibald} im Bull. Amer. Math. Soc. (2) 19, 421-422. Es soll an die Stelle der Notions de math茅matiques von \textit{J. Tannery} treten, das wir nur mit dem Titel anf眉hren konnten (F. d. M. 34, 80 (JFM 34.0080.*), 1903), gerade wie seine deutsche 脺bersetzung (F. d. M. 40, 227 (JFM 40.0227.*) u. 541, 1909). Obschon das neue Werk ``viel gr枚脽er ist, sind die behandelten Gegenst盲nde nicht so zahlreich, und die ganze Behandlungsweise ist gr眉ndlich verschieden. An den meisten Universit盲ten Frankreichs wird den Studenten der Physik, Chemie und Technik ein Lehrgang in allgemeiner Mathematik geboten. Algebra, Analysis und Mechanik werden in ihm entwickelt. Als eine Vorbereitung f眉r solche Kurse und zur Ausf眉llung der hiermit zusammenh盲ngenden L眉cken ist das vorliegende Buch abgefa脽t. W盲hrend die Strenge nicht vernachl盲ssigt ist, h盲lt sich die Darstellung nicht bei Einzelheiten der Beweise auf, und es werden praktische Anwendungen der verschiedenen Gegenst盲nde betont.''. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Let \(U_ q\) denote the quantized enveloping algebra over \({\mathbb{Q}}(q)\) associated to a symmetrizable Kac-Moody algebra \({\mathfrak g}\). For any integrable \(U_ q\)-module M the author defines a crystal base for M to be a pair (L,B) consisting of a lattice L of M and a \({\mathbb{Q}}\)-basis B of L/qL with certain nice properties. In the paper under review the author proves the existence and uniqueness of crystal bases for the case where \({\mathfrak g}\) is a finite dimensional classical Lie algebra, and in the paper reviewed above he announces the extension to the general case [see also Preprint 728, Res. Inst. Math. Sci., Kyoto Univ. for details and proofs)].
\textit{G. Lusztig} has constructed a so-called canonical basis for the \(+\) part of \(U_ q\) (for types A, D and E), see [J. Am. Math. Soc. 3, 447- 498 (1990; Zbl 0703.17008) and Prog. Theor. Phys. Suppl. 102, 175-201 (1990)]. On the irreducible integrable highest weight modules the two constructions lead to the same bases. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The Schwarz alternating method has been used to find the second-order approximation to the problem of the uniform compression of an elastic body containing two equally-sized circular holes. The solution is extremely accurate for holes whose centers are not closer than about 1.10 diameters. The approximate solution obtained is actually much more general than may appear, since, for example, the case of two holes of different diameters can be treated by merely making the correct change of variables in equations. Furthermore, the second-order approximation for the compression of a body with any number of holes can be written as a series of ``two-hole'' solutions. Such approximate solutions may be of use in estimating the effective elastic moduli of porous materials. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Verf. macht drei Voraussetzungen, bzw. Annahmen: 1. Das Relativit盲tsprinzip, da脽 f眉r alle Koordinatensysteme, f眉r welche die mechanischen Gleichungen gelten, auch die gleichen elektrodynamischen und optischen Gesetze gelten. 2. Die Unabh盲ngigkeit der Lichtgeschwindigkeit von der Geschwindigkeit des emittierenden K枚rpers. 3. Die Definition der gemeinsamen Zeit f眉r zwei synchron gehende Uhren, wonach die ``Zeit'', die das Licht beim Hingange von der einen zur anderen Uhr braucht, gleich sein soll der Zeit des R眉ckganges. Unter diesen Voraussetzungen kommt Verf. zu einer allgemeinen Theorie der Relativit盲t, die f眉r die Elektrodynamik im wesentlichen zu der \textit{Lorentz}schen Theorie mit ihren neueren Erg盲nzungen f眉hrt. Die Transformationen, die auf die Gleichungen der Elektrodynamik in einem bewegten System anzuwenden sind, lauten:
\[
\begin{aligned} \tau= & \beta\left(t-\frac{v}{V^2}\,x\right),\\ \xi = & \beta(x-vt),\quad \eta=y,\quad \zeta=z,\\ & \beta = \frac{1}{\sqrt{1-(\frac vV)^2}}\,,\end{aligned}
\]
wobei \(v\) die Geschwindigkeit des bewegten Systems bedeutet. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The purpose of this paper is to present the inverse obstacle scattering problems for time-harmonic waves by considering the case of two dimensions, scattering of time-harmonic acoustic or electromagnetic waves by infinitely long cylindrical obstacles. For the approximation of the solution of the inverse problem is considered a regularized Newton method. The numerical schemes are based on the solution of boundary integral equations by the Nystr枚m method. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The author introduces a new polynomial invariant \(V_ L(t)\) for tame oriented links via certain representations of the braid group. That the invariant depends only on the closed braid is a direct consequence of Markov's theorem and a certain trace formula, which was discovered because of the uniqueness of the trace on certain von Neumann algebras. There is an alternate way to calculate \(V_ L(t)\) without converting L into a closed braid, using only a Conway type relation; \(V_{unknot}=1\) and 1/t \(V_{L-}-t V_{L+}=(\sqrt{t}-1/\sqrt{t})V_ L\). This is also interesting from a view point of formal knot theory. The author gives many results using this invariant. For an example, \(V_{L\sim}(t)=V_ L(1/t)\) where \(L\sim\) means the mirror image of L, \(V_{L_ 1\#L_ 2}=V_{L_ 1}\cdot V_{L_ 2}\) where {\#} means a connected sum of links, \(V_ L(-1)=\Delta_ L(-1)\) where \(\Delta_ L\) means the Alexander polynomial, \(V_ L(1)=(-2)^{p-1}\) where p is the number of components of L, \(V_ K(e^{2\pi i/3})=1\) and d/dt \(V_ K(1)=0\) if K is a knot. If K is a knot and \(| \Delta_ K(i)| >3\), then k cannot be represented as a closed 3 braid. If K is a knot and \(\Delta (e^{2\pi i/5})>6.5\), then K cannot be represented as a closed 4 braid. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Young infants learn words by detecting patterns in the speech signal and by associating these patterns to stimuli provided by non-speech modalities (such as vision). In this paper, we discuss a computational model that is able to detect and build word-like representations on the basis of multimodal input data. Learning of words (and word-like entities) takes place within a communicative loop between a `carer' and the `learner'. Experiments carried out on three different European languages (Finnish, Swedish, and Dutch) show that a robust word representation can be learned in using approximately 50 acoustic tokens (examples) of that word. The model is inspired by the memory structure that is assumed functional for human speech processing. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The author constructs the global crystal bases of the \(q\)-analogue \(A_ q({\mathfrak g})\) of the coordinate ring of the reductive algebraic group associated with the Lie algebra \({\mathfrak g}\) by using the same method as in his paper [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)]. The main theorem of this paper is in \S7.4:
Theorem 1. (i) \(A_ q^{\mathbb{Q}}({\mathfrak g})\cap L(A_ q({\mathfrak g}))\cap \overline{L}(A_ q({\mathfrak g})) \to L(A_ q({\mathfrak g}))/qL(A_ q({\mathfrak g}))\) is an isomorphism. (ii) Letting \(G\) be the inverse of the isomorphism above, we have \(A_ q^{\mathbb{Q}}({\mathfrak g})= \oplus_{b\in B(A_ q({\mathfrak g}))} \mathbb{Q}[q,q^{-1}] G(b)\). The author also gives an explicit form of the global crystal base of \(A_ q(sl_ 2)\) and examines Berenstein and Zelevinsky's conjecture as an example. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Der erste Teil der Arbeit behandelt allgemeine Tetraeder. S盲tze von \textit{Painvin} und \textit{Beltrami} 眉ber eine die Seiten des Tetraeders enthaltende Fl盲che dritter Ordnung ergeben durch Spezialisierung den Satz von \textit{Sartiaux}: F盲llt man von der Mitte einer Geraden, die zwei von den Zentren der acht Ber眉hrungskugeln eines Tetraeders verbindet, die Lote auf die Tetraederfl盲chen so liegen ihre Fu脽punkte in einer Ebene.'' Hierzu wird ein elementar-geometrischer Beweis gegeben, dem sich noch einige Betrachtungen anschlie脽en.
An der Spitze des zweiten Abschnitts, der das Dreieck in der ebenen Geometrie betrifft, steht der \textit{Steiner-Faure}sche Satz: ``Die Bedingung daf眉r, da脽 sich Poldreiecke eines Kegelschnitts \(V\) einem Kreise \(U\) einschreiben oder Poldreiecke des Kreises \(U\) dem Kegelschnitt \(V\) umschreiben lassen, ist, da脽 \(U\) den Direktorkreis von \(V\) rechtwinklig schneidet. ``Der Satz f眉hrt zur Aufstellung der Formeln:
\[
\overline{OH}^2=R^2+2(H), \overline{IH}^2=2r^2+(H),
\]
wo \(O\) und \(I\) die Mittelpunkte von Um- und Inkreis, \(R\) und \(r\) ihre Radien, \(H\) den H枚henschnittpunkt und \((H)\) die Potenz desselben in bezug auf den Umkreis (im Fall eines stumpfwinkligen Dreiecks das Quadrat des Radius des sogenannten konjugierten Kreises) bedeutet. Herangezogen wird auch der \textit{Feuerbach}sche Kreis und der Kreis mit dem Durchmesser \(GH\), wobei \(G\) der Schwerpunkt ist.
Der dritte Abschnitt beginnt mit der Aufstellung analoger Formeln f眉r das orthozentrische Tetraeder:
\[
\overline{OH}^2=R^2+3(H), \overline{HI}^2=3r^2+(H).
\]
Sie werden zur Herleitung des folgenden bemerkenswerten Satzes verwendet: ``Liegt von zwei Kugeln \((O,R)\) und \((I,r)\) die zweite innerhalb der ersten, so existieren \(\infty^2\) orthozentrische Tetraeder, die der ersten Kugel ein-, der zweiten umgeschrieben sind.'' F眉r die ausgezeichneten Punkte dieser Tetraeder werden die geometrischen 脰rter bestimmt.
Im letzten Teile der Arbeit werden f眉r das orthozentrische Tetraeder 眉ngleichheiten entwickelt, die 眉ber verschiedene Lagebeziehungen Aufschlu脽 geben. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The authors construct the generalization of the Jones polynomial of links to the case of graphs in \(R^ 3\). They introduce the so-called coloured ribbon graphs in \(R^ 2 \times [0,1]\) and define for them Jones-type isotopy invariants. The approach to colouring is based on Drinfeld's notion of a quasitriangular Hopf algebra. For each quasitriangular Hopf algebra \(A\) the authors define \(A\)-coloured (ribbon) graphs. The colour of an edge is an \(A\)-module. The colour of a vertex is an \(A\)-linear homomorphism intertwining the modules which correspond to edges incident to this vertex. The category of an \(A\)-coloured ribbon graph is a compact braided strict monoidal category introduced by \textit{A. Joyal} and \textit{R. Street} [Braided monoidal categories, Macquarie Math. Reports, Report No. 860081 (1986)]. If \(A\) satisfies a minor additional condition, then the authors construct a canonical covariant functor from the category of \(A\)- coloured ribbon graphs into the category of \(A\)-modules. In the case of \(A\) being the quantized universal enveloping algebra of \(sl_ 2\) this functor generalizes the Jones polynomial of links. If \(A=U_ h(sl_ n(C))\) this generalizes the Jones-Conway (Thomflyp) polynomial and for \(A=U_ hG\), \(G=so(n)\), \(sp(2k)\) the Kauffman polynomial. The paper under review was followed by the authors' paper ``Invariants of 3-manifolds via link polynomials and quantum groups'' [Invent. Math. 103, 547-597 (1991; Zbl 0725.57007)]in which the authors construct new topological invariants of compact oriented manifolds (employing the methods of the paper under review). The construction was partially inspired by ideas of \textit{E. Witten} [Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351-399 (1989; Zbl 0667.57005)]who considered quantum field theory defined by the nonabelian Chern-Simon action and applied it to the study of 3-manifolds (on physical level of rigor). | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The sage-drg package for the SageMath computer algebra system has been originally developed for computation of parameters of distance-regular graphs. Recently, its functionality has been extended to handle general association schemes. The package has been used to obtain nonexistence results for both distance-regular graphs and \(Q\)-polynomial association schemes, mostly using the triple intersection numbers technique. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Das au脽erordentliche Interesse an Stringtheorien beruht darauf, da脽 sie erstmals eine konsistente einheitliche quantentheoretische Beschreibung aller fundamentalen Teilchen und deren Wechselwirkungen inklusive der Gravitation als m枚glich erscheinen lassen und da脽 sich dabei Strukturen offenbaren, deren Untersuchung auch mathematisch bedeutsam ist.
Verff. haben mit ihrem Buch ''Superstring Theory'' eine gut lesbare, systematische und p盲dagogische Darstellung des bis zum Jahre 1986 vorhandenen und von ihnen mit Kompetenz ausgew盲hlten aussageg眉ltigen Materials vorgelegt. Der Umfang von mehr als eintausend Seiten, die auf zwei B盲nde mit jeweils acht Kapiteln verteilt sind, weist auf noch fehlende Prinzipien hin, die den String letztlich definieren. Es m眉ssen verschiedene Zug盲nge zur selben Problematik parallel behandelt werden, um nicht wesentliche Aussagen f眉r das gegenw盲rtige Verst盲ndnis einer Stringtheorie zu verlieren.
So wichtige Problemkreise wie die String-Feldtheorien oder Polyakovs Wegintegral-Formulierung von Strings in nichtkritischen Raum-Zeit- Dimensionen m眉ssen dabei unber眉cksichtigt bleiben, zum Teil auch wegen der Unbestimmtheit ihrer Aussagen. Oft sind Themen eingestreut, die nur schematisch behandelt werden, die aber dem Verst盲ndnis der Gesamtproblematik dienen. Gerade hier sto脽en wir auf ungel枚ste Probleme, z.B. die Funktionalintegration 眉ber Riemannsche Fl盲chen mit h枚herem Genus, die die Stringwechselwirkung h枚herer Ordnungen definiert.
Der erste Band, der nach einem allgemein einf眉hrenden Kapitel die Quantisierung freier Strings mit unterschiedlichen inneren und 盲u脽eren (Super-)Symmetrien und die Wechselwirkung bis zur ''tree- approximation'' behandelt, setzt nur geringe mathematische und quantenfeldtheoretische Kenntnisse voraus. Es kommen haupts盲chlich Fockraum-Methoden und selten die Wegintegration zur Anwendung. Die Konstruktion nichtabelscher heterotischer Superstring-Theorien erfordern Torus-Kompaktifizierung und die Vertex-Operator-Darstellung affiner Liealgebren. Unendlichdimensionale Virasoro-Algebren, die hier zur Charakterisierung der Diffeomorphismen, die Symmetrietransformationen aller Stringtheorien sind, ausreichen, m眉ssen f眉r die im zweiten Band besprochenen Wechselwirkungen h枚herer Ordnung durch globale Diffeomorphismen erzeugende modulare Transformationen erg盲nzt werden.
Zur Vermeidung konformer und Gravitations-Anomalien wird im ganzen Buch generell mit krititschen Raum-Zeit-Dimensionen gearbeitet. Der vom Verst盲ndnis her anspruchsvollere, in den letzten Kapiteln aber spekulative zweite Band, in dem 1-Loop-Amplituden, Eichanomalien, effektive Niederenergie-Feldtheorien und deren Raum-Zeit- Kompaktifizierung dargestellt sind, enth盲lt Kapitel 眉ber Differentialgeometrie und algebraische Geometrie, die f眉r die Kompaktifizierung und sicher auch f眉r die Aufkl盲rung der Struktur Riemannscher Fl盲chen wichtig sind.
Hier ist es angebracht zu betonen, da脽 das vorliegende Buch von Physikern und f眉r diese geschrieben wurde. Es ist jedoch f眉r einen weit gr枚脽eren Leserkreis geeignet, f眉r alle, die Anregung durch Stringtheorien und Orientierung auf diesem ausufernden Gebiet suchen, weil f眉r jedes Kapitel am Schlu脽 eines jeden Bandes eine nach historischen Gesichtspunkten geordnete Diskussion der wichtigsten Ideen mit zahlreichen Literaturzitaten beigef眉gt ist. Dabei f盲llt der String-Feldtheorie ein eigener Abschnitt zu. Dem engagiert und ideenreich geschriebenen Buch ist eine gro脽e Leserschaft zu w眉nschen. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. We study thermodynamics of the parabolic Lemaitre-Tolman-Bondi (LTB) cosmology supported by a perfect fluid source. This model is the natural generalization of the flat Friedmann-Robertson-Walker (FRW) universe, and describes an inhomogeneous universe with spherical symmetry. After reviewing some basic equations in the parabolic LTB cosmology, we obtain a relation for the deceleration parameter in this model. We also obtain a condition for which the universe undergoes an accelerating phase at the present time. We use the first law of thermodynamics on the apparent horizon together with the Einstein field equations to get a relation for the apparent horizon entropy in LTB cosmology. We find out that in LTB model of cosmology, the apparent horizon's entropy could be feeded by a term, which incorporates the effects of the inhomogeneity. We consider this result and get a relation for the total entropy evolution, which is used to examine the generalized second law of thermodynamics for an accelerating universe. We also verify the validity of the second law and the generalized second law of thermodynamics for a universe filled with some kinds of matters bounded by the event horizon in the framework of the parabolic LTB model. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The paper represents an expanded version of a lecture at the IAMP Congress, Swansea, July 1988. It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three-dimensional terms. In this version, the Jones polynomial can be generalized from \(S^ 3\) to arbitrary manifolds that are computable from a surgery presentation. These results shed new light on conformal field theory in \(1 + 1\) dimensions, which can be generated essentially by studying the generally covariant \(2 + 1\) dimensional theory on various three manifolds with boundary. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Artificial Neural Networks (ANN) deal with information through interactions among neurons (or nodes), approximating the mapping between inputs and outputs based on nonlinear functional composition. They have the advantages of self-learning, self-organizing, and self-adapting. It is practical to use ANN technology to carry out hydrologic calculations. To this end, this note has fundamentally set up a system of calculation and analysis based on ANN technology, and given an example of application with good results. It shows that ANN technology provides a relatively effective way of solving problems in hydrologic calculation. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Contents: Jean B茅nabou, Introduction to bicategories (1--77); A. Dold, S. MacLane und U. Oberst, Projective classes and acyclic models (78--91); Robert Davis, Equational systems of functors (92--109); John R. Isbell, Normal completions of categories (110--155); Jan-Erik Roos, Locally distributive spectral categories and strongly regular rings (156--181).
Extensive reviews of the individual articles are given in scanned review text:
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\url{http://zbmath.org/scans/165/334.gif}. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. We consider an efficient handling of matrices arising in surface interpolation and approximation with radial basis functions. To find a data-sparse approximation of the system matrix, the adaptive cross-approximation technique is used. The approximation of the matrix requires \(O (N\log^{2} N)\) units of storage and arithmetic operations, where \(N\) is the number of interpolation points. Because basis functions are not explicitly used, the implementation is applicable to a wide class of interpolation kernels. We present numerical examples involving generated data and measurements of formed sheet-metal parts. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The author associates to each diagram \(D\) of a link \(L\) in three-space a chain complex \(C(D)\) of graded \(Z[c]\)-modules, where \(c\) is an indeterminate of degree 2. The graded cohomology of \(C(D)\) is a link type invariant of \(L\) whose graded Euler characteristic determines the Kauffman bracket, and hence the Jones polynomial, of \(L\). | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. We study the robustness of biological systems by means of the eigenstructure of the deviation curvature tensor. This is the differential geometric theory of variational equations for deviations of whole trajectories to nearby ones. We apply this theory to Van der Pohl equations and some biological models, and examine the relationship between the linear stability of steady-states and the stability of transient states. The main application is the \(G_1\)-model for the cell cycle, where Jacobi stability reveals the robustness and fragility of the cell arrest states and suggests the existence of more subtle checkpoints. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. [For the entire collection see Zbl 0646.00009.]
A category \({\mathcal C}\) is introduced with a sequence \(C_0, C_1, C_2,\ldots,\). where \(C_n\) is a disjoint union of \(n\) oriented circles. A morphism \(C_n\to C_m\) is a Riemann surface bounded by the corresponding circles (which geometrically looks like a surface connecting \(n\) circles with \(m\) circles by a surface -- or connecting \(n\) closed strings with \(m\) closed strings -- in physical application to string theory). The conformal field theory in dimension 2 (one space and one time dimension), which is related to string theory, is then defined as a representation of \({\mathcal C}\) in a complex Hilbert space. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. We present a brief discussion of some basic features of HIV dynamics. Within a multiple compartment framework, we model the dynamics of HIV subsequent to the application of a highly active antiretroviral therapy (HAART) considering blood and lymphatic system compartments. We show that T-cell and HIV decay rates after highly active antiretroviral therapy (HAART) correspond to time dependent effective coefficients which include transfer between the compartments, stressing that a main component in the evolution of viral concentration is redistribution.
In our analysis the viremia decay rate appears to be mainly determined by the effective lymph node viral decay rate rather than by the decay constant of infected T CD4 cells. We also show that in a two type T cell model with a transition from type 1 to type 2, the more active cells with a shorter life are the progenitors of the less active ones. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Using techniques of quantum field theory, Witten defined, on a physical level of rigor, a series of new invariants of 3-manifolds and of links in 3-manifolds. The present paper provides a mathematical background for the construction of these invariants. For closed 3-manifolds, they form a sequence of complex numbers parametrized by complex roots of unity which specialize to the values of the Jones polynomial at the corresponding roots of unity for a link in the 3-sphere. For manifolds with boundary they form a sequence of finite dimensional complex linear operators producing for each root of unity a 3-dimensional topological quantum field theory. The paper uses the algebraic language of Hopf algebras. A central ingredient of the construction of the new invariants are the Kirby-moves relating different surgery presentations by framed links in the 3-manifold. Certain invariants of framed links in the 3-sphere combine in expressions invariant under these Kirby-moves thus giving the new invariants. The basic invariant used is the Jones polynomial which is known to be connected with the quantum enveloping algebra of the Lie algebra \(sl_ 2({\mathbb{C}})\) and its fundamental representation; other irreducible representations of the algebra are used to construct more basic invariants of links in the 3-sphere, which serve then as a groundstock for the construction of the invariants of general 3- manifolds.
A purely topological and combinatorial construction of the invariants, thus avoiding the algebraic machinery of the present paper, can be found in papers by \textit{W. B. R. Lickorish} [Pac. J. Math. 149, No.2, 337-347 (1991)] and \textit{K. H. Ko} and \textit{L. Smolinsky} [ibid., 319-336 (1991)]. Concrete computations or applications of the new invariants have not yet appeared. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Although complete randomization ensures covariate balance on average, the chance of observing significant differences between treatment and control covariate distributions increases with many covariates. Rerandomization discards randomizations that do not satisfy a predetermined covariate balance criterion, generally resulting in better covariate balance and more precise estimates of causal effects. Previous theory has derived finite sample theory for rerandomization under the assumptions of equal treatment group sizes, Gaussian covariate and outcome distributions, or additive causal effects, but not for the general sampling distribution of the difference-in-means estimator for the average causal effect. We develop asymptotic theory for rerandomization without these assumptions, which reveals a non-Gaussian asymptotic distribution for this estimator, specifically a linear combination of a Gaussian random variable and truncated Gaussian random variables. This distribution follows because rerandomization affects only the projection of potential outcomes onto the covariate space but does not affect the corresponding orthogonal residuals. We demonstrate that, compared with complete randomization, rerandomization reduces the asymptotic quantile ranges of the difference-in-means estimator. Moreover, our work constructs accurate large-sample confidence intervals for the average causal effect. | 0 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. We derive (quasi-)quantum groups in \(2+1\) dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in a \(d+1\) dimensional topological theory to manifolds of dimension less than \(d+1\). We then ``construct'' a generalized path integral which in \(d+1\) dimensions reduces to the standard one and in \(d\) dimensions reproduces the quantum Hilbert space. In a \(2+1\) dimensional topological theory the path integral over the circle is the category of representations of a quasi- quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here. | 1 |
As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective.
The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincar茅}'s fundamental groups [J. de l'脡c. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincar茅} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. G枚ttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schr枚dinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schr枚dinger found out his famous equation to be called the Schr枚dinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. W眉thrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. B茅nabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized
about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes 脡tud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their
appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The
forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z眉rich, Switzerland. Vol. I. Basel: Birkh盲user. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)].
The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every
effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. For a special family of hypoelliptic operators containing the semielliptic operators the functional characteristic of hypoellipticity is introduced, which is equivalent to hypoellipticity index in the elliptic case and sharpens it in the general case. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. A quantum field theory of Einstein's general relativity is formulated in indefinite-metric Hilbert space in such a way that asymptotic fields are manifestly Lorentz covariant and the physical \(S\)-matrix is unitary. The general coordinate transformation is transcribed into a \(q\)-number transformation, called the Becchi-Rouet-Stora (BRS) transformation. Its abstract definition is presented on the basis of the BRS transformation for the Yang-Mills theory. The BRS transformation for general relativity is then explicitly constructed. The gauge-fixing Lagrangian density and the Faddeev-Popov (FP) one are introduced in such a way that their sum behaves like a scalar density under the BRS transformation. One can then proceed in the same way as in the Kugo-Ojima formalism of the Yang-Mills theory to establish the unitarity of the physical \(S\)-matrix. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. It has become increasingly possible for mathematics students to include a limited number of approved optional, non-traditional modules in their programs. Surveys coursework in four non-specialist modules over a seven-year period, and examines work in the area of mathematical cognition. (Contains 16 references.) (ERIC) | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Dieser Artikel erschien in dem im Zbl 0213.00103 angezeigten Sammelwerk.]
Aus der Einleitung des Verf.: ``The \(\ell\)-adic 茅tale cohomology of algebraic varieties is much richer than the classical cohomology, in that Galois groups operate on it. This opens up a new field of inquiry, even in the classical case. Although theorems seem scarce, the soil is fertile for conjectures\(\ldots\). The main idea is, roughly speaking, that a cohomology class which is fixed under the Galois group should be algebraic when the ground field is finitely generated over the prime field.''
In den ersten beiden Paragraphen (\S1. The \(\ell\)-adic cohomology; \S2. Cohomology classes of algebraic cycles) wird diese Idee n盲her ausgef眉hrt. Sei \(k\) ein K枚rper, \(\bar k\) eine algebraisch abgeschlossene Erweiterung von \(k\), \(G(\bar k,k)\) die Automorphismengruppe von \(\bar k,k\). Sei \(V\) ein geometrisch irreduzibles und glattes projektives Schema 眉ber \(k\), sei \(\bar V=V\times\bar k\). Auf den \(\ell\)-adischen Kohomologiegruppen \(H_{\ell}^i(\bar V)\) operiert \(G(\bar k,k)\); um die ,,richtigen \(\ell\)-adischen Orientierungen'' zu erhalten, werden ,,tordierte'' Kohomologiegruppen \(H_{\ell}^i(\bar V)(m)\) eingef眉hrt. Einem irreduziblen Unterschema \(X\) von \(\bar V\) der Kodimension \(i\) wird eine Kohomologieklasse \(c(X)\in H_{\ell}^{2i}(\bar V)(i)\) zugeordnet. \(X\) ist ,,definiert'' 眉ber einer endlichen Erweiterung von \(k\) und wird daher durch eine offene Untergruppe \(U\) von \(G(\bar k,k)\) festgelassen. Dann wird auch \(c(X)\) durch \(U\) festgelassen. Die erste Vermutung des Verf. besteht darin, da脽 die Umkehrung dieser Tatsache richtig ist (falls \(k\) 眉ber dem Primk枚rper endlich erzeugt wird). Aus dieser Vermutung ergibt sich u.a. fur Abelsche Mannigfaltigkeiten eine Konsequenz (Formel (8)), die inzwischen vom Verf. selbst bewiesen wurde (falls \(k\) endlicher K枚rper ist; s. die Arbeit in Invent. Math. 2, 134--144 (1966; Zbl 0147.20303). Gegenw盲rtig gibt es auch weitere Information 眉ber die in \S1 betrachteten(von Serre eingef眉hrten) Lieschen Algebren. Siehe \textit{J.-P. Serre} [Abelian \(\ell\)-adic representations and elliptic curves. New York etc.: Benjamin (1968; Zbl 0186.25701)].
In \S3 diskutiert Verf. den Zusammenhang seiner Vermutungen mit den Zetafunktionen von Variet盲ten 眉ber endlichen K枚rpern. Im Mittelpunkt steht die Vermutung, da脽 die Ordnung des Pols der Zetafunktion \(\zeta(V,s)\) an der Stelle \(s=i\) gleich dem Rang der durch die algebraischen Zyklen der Kodimension \(i\) erzeugten Gruppe ist (Formel (12)).
In \S4 wird nur vorausgesetzt, da脽 \(k\) 眉ber seinem Primkorper endlich erzeugt ist. Verf. kombiniert seine Vermutungen mit den bekannten Hypothesen von Birch und Swinnerton-Dyer uber die Zetafunktion von \(V/k\). Ein angek眉ndigtes Ergebnis von \textit{H. Pohlmann} ist inzwischen publiziert [Ann. Math. (2) 88, 161--180 (1968; Zbl 0201.23201)].
Weitere erg盲nzende bibliographische Angaben:
\textit{A. Grothendieck} [``Standard conjectures on algebraic cycles.'' Algebr. Geom., Bombay Colloq. 1968, 193--199 (1969; Zbl 0201.23301)];
\textit{P. Swinnerton-Dyer} [``The conjectures of Birch and Swinnerton-Dyer, and of Tate.'' Proc. Conf. local Fields, NUFFIC Summer School Driebergen 1966, 132--157 (1967; Zbl 0197.47101)];
the author [``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog.'' S茅m. Bourbaki 1965/66, Exp. No. 306, 415--440 (1966; Zbl 0199.55604)];
\textit{J.-P. Serre} [S茅m. Delange-Pisot-Poitou 11 (1969/70), Exp. No. 19 (1970; Zbl 0214.48403)]. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. An operator triple system means a complex linear subspace of \({\mathcal L}(H,K)\), the bounded linear operators from a Hilbert space H to a Hilbert space K, which is closed in some topology and under some triple product \(\{\).,.,.\(\}\) of its elements. Examples of operator triple systems are given by \(J^*\)-algebras, norm closed subspaces of \({\mathcal L}(H,K)\) which contain \(aa^*a\) along with the element a; this is because a \(J^*\)-algebra is closed under the triple product \(\{a,b,c\}=ab^*c+cb^*a.\) This paper deals with triple systems which satisfy the identities
\[
\{xy\{zuv\}\}=\{x\{uzy\}v\}=\{\{xyz\}uv\},
\]
i.e. associative Jordan triple systems. In particular if a \(J^*\)- algebra becomes an associative Jordan triple system in the triple product \(\{xyz\}=1/2(xy^*z+zy^*x)\) then it is said to be a commutative \(J^*\)-algebra.
The main result of the paper (Theorem 1) states that a commutative \(J^*\)-algebra is isometrically \(J^*\)-isomorphic to a space of all continuous complex functions satisfying some symmetry properties. One of the principal consequences of the main result is a Gelfand-Neumark representation theorem for associative Jordan triple systems (Theorem 2). There are also some applications of Theorem 1 to the study of contractive projections, Banach-Stone type theorems, Stone-Weierstrass theorems in the setting of commutative \(J^*\)-algebras. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. A generalization procedure which reconstructs mathematical inductions that are expanded in a proof of an example is formulated under a typed \(\lambda\)-calculus that is newly defined for increasing the applicability of the procedure. The \(\lambda\)-calculus, an extension of logical framework, allows recursions and inductions on natural numbers, and inferences on linear arithmetical terms are built into its type system. The generalization procedure is iterated in a bottom-up fashion to construct nested inductions. Consequently, it can also find inductions whose induction formula is a limited form of bounded quantification. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We present, in a simplified setting, a non-commutative version of the well-known Gel'fand-Na沫mark duality (between the categories of compact Hausdorff topological spaces and commutative unital \(C^*\)-algebras), where ``geometric spectra'' consist of suitable finite bundles of one-dimensional \(C^*\)-categories equipped with a transition amplitude structure satisfying saturation conditions. Although this discrete duality actually describes the trivial case of finite-dimensional \(C^*\)-algebras, the structures are here developed at a level of generality adequate for the formulation of a general topological/uniform Gel'fand-Na沫mark duality, fully addressed in a companion work. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ 2\) or \({\mathbb{R}}^ 3\) with a smooth boundary. We consider an inhomogeneous viscous incompressible fluid occupying \(\Omega\). Let \(\rho\),u,p be the mass density, the velocity vector and the pressure of the fluid, respectively. Then these quantities obey the following system of equations
\[
\partial \rho /\partial t+u\cdot \nabla \rho =0\quad (0<t,\quad x\in \Omega),
\]
\[
\rho \{\partial u/\partial t+(u\cdot \nabla)u\}=\Delta u-\nabla p\quad (0<t,\quad x\in \Omega),
\]
\[
div u=0\quad (0<t,\quad x\in \Omega).
\]
This system is a generalization of the Navier-Stokes system.
We employ the \(L^ 2\)-theory and prove the unique existence (local in time) of the solution. The main tool is the theory of linear evolution equations. This local existence theorem is used to prove the following global existence theorem:
I) In the two-dimensional problem the solution always exists globally in time.
II) In the three-dimensional problem the global solution is obtained, if the initial values are sufficiently small.
The important fact is that in the two-dimensional problem we do not need to assume smallness of the initial values.
This paper consists of seven sections. In section 2 we give various function spaces and we formulate an initial boundary value problem in the framework of the theory of evolution equations. Main theorems are also stated in this section. Sections 3, 4 and 5 deal with the proof of the local existence theorem. The global existence of the solution of the two- dimensional problem is proved in section 6. The three-dimensional case is considered in section 7. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We propose an MLP/HMM hybrid model in which the input feature vectors are transformed by nonlinear predictors using MLPs assigned to each state of a HMM. The prediction error vectors in the states are modeled by Gaussian mixture densities. The use of a hybrid model is motivated from the need to model the prediction errors in the conventional neural prediction model where the prediction errors are variable due to the effect of varying contexts and speaker identity. The hybrid model is advantageous because frame-correlation in the input speech signal is exploited by employing the MLP predictors, and the variabilities in the prediction error signals are explicitly modeled. We present the training algorithms based on the maximum likelihood criterion and discriminative criterion for minimum error classification. Experiments were done on speaker-independent continuous speech recognition. (Provider: Leibiger) | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Diese Vorlesungsausarbeitung ist die erste zusammenfassende Darstellung der Satoschen Theorie der Mikrofunktionen und der Folgerungen f眉r partielle Differentialgleichungen, die daraus von den drei Verff. abgeleitet wurden. Die Lekt眉re ist jedoch nicht einfach. Sei \(M\) eine reell-analytische Mannigfaltigkeit, \(S^*M\) das Cosph盲renb眉ndel 眉ber \(M\), \(\mathcal A\) die Garbe der Keime analytischer Funktionen und \(\mathcal B\) die Garbe der Hyperfunktionskeime auf \(M\). Eine Mikrofunktion ist ein Schnitt der Garbe \(\mathcal C\) auf \(S^*M\), die die Eigenschaft hat, dass \(\pi^*\mathcal C\cong \mathcal B/\mathcal A\) ist, dabei ist \(\pi\) die kanonische Projektion von \(S^*M\) auf \(M\). Die Mikrofunktionen liefern eine spektrale Zerlegung der Singularit盲ten von Hyperfunktionen. F眉r Distributionen findet man eine 盲hnliche Konstruktion bei \textit{L. H枚rmander} [Acta Math. 127, 79--183 (1971; Zbl 0212.46601)]. Die Konstruktion der Garbe \(\mathcal C\) verwendet wesentlich kohomologische Methoden, wie relative Kohomologiegruppen, die derivierte Kategorie und das Verschwinden gewisser Kohomologiegruppen und ist zu kompliziert, um hier wiedergegeben zu werden. Eine elementare Konstruktion findet man bei \textit{P. Schapira} [in: S茅min. Goulaouic-Schwartz 1970-脡quat. d茅riv. part. Analyse fonction. No. 11, 15 p. (1971; Zbl 0234.46044)]. Mit Hilfe der Garbe \(\mathcal C\) wird der wesentliche Tr盲ger \(SS(u)\) einer Hyperfunktion \(u\) definiert. (Analog daf眉r ist f眉r eine Distribution \(u\) die Wellenfront \(WF(u)\); vgl. [Zbl 0212.46601]). Ist \((X,\xi)\) ein Punkt in \(S^*M\) (in lokalen Koordinaten) so ist \((X,\xi)\not\in SS(u)\), wenn \(u\) in einer Umgebung \(\Omega\) von \(x\) sich als endliche Summe \(u=\sum_{j=1}^k u_j\) schreiben l盲sst; dabei ist \(u_j\) Randwert einer Funktion, die in \(\Omega+i\Gamma_j\) holomorph ist, und die \(\Gamma_j\), \(j=1,\dots,k\), sind Kegel, die in \(\{y:y\in\mathbb{R}^n,\langley,\xi\rangle<0\}\) enthalten sind. Die Projektion von \(SS(u)\) auf \(M\) ist der singul盲re Tr盲ger von \(u\), d.h. die Menge der Punkte in \(M\), in denen \(u\) nicht reell-analytisch ist. Es werden dann Beispiele angegeben und das Rechnen mit Mikrofunktionen behandelt. Als Folgerung ergibt sich der Satz von Sato, dass f眉r due L枚sungen der Gleichung \(Pu=0\), \(P\) ein Pseudodifferentialoperator (endlicher Ordnung) gilt: \(SS(u)\subset\{(x,\xi):P_m(x,\xi)=0\}\), wobei \(P_m(x,\xi)\) das Hauptsymbol von von \(P\) ist. Weiter wird die Struktur von Systemen von Pseudodifferentialoperatoren untersucht. Es zeigt sich, dass unter gewissen Voraussetzungen ein System von Pseudodifferentialoperatoren mikrolokal (d.h. lokal bez眉glich \(x\) und \(\xi\)) mittels kanonischer Transformationen auf eine einfache Form transformiert werden kann, die sich grob gesprochen aus einigen einfachen Typen von Differentialoperatoren (Cauchy-Riemann-System, de Rham-System, Gleichung von hans Lewy) zusammensetzt. Daraus ergeben sich Folgerungen f眉r die L枚sungen. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The authors resolve the question that finding a Nash equilibrium in a three players game is indeed PPAD-complete which is established by a solid theoretical and mathematical foundation. A number of propositions, lemmas, theorems are proposed and presented with some illustration. Some of the algorithms developed are not performed for illustration. May be the authors will take it up in subsequent research papers. It is an interesting paper for researchers in the area of game theory. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Inter-universal Teichm眉ller theory may be described as a sort of arithmetic version of Teichm眉ller theory that concerns a certain type of canonical deformation associated to an elliptic curve over a number field and a prime number \(l\leq 5\). We begin our survey of interuniversal Teichm眉ller theory with a review of the technical difficulties that arise in applying scheme-theoretic Hodge-Arakelov theory to diophantine geometry. It is precisely the goal of overcoming these technical difficulties that motivated the author to construct the nonscheme-theoretic deformations that form the content of inter-universal Teichm眉ller theory.
Next, we discuss generalities concerning ``Teichm眉ller-theoretic deformations'' of various familiar geometric and arithmetic objects which at first glance appear one-dimensional, but in fact have two underlying dimensions. We then proceed to discuss in some detail the various components of the log-theta-lattice, which forms the central stage for the various constructions of inter-universal Teichm眉ller theory. Many of these constructions may be understood to a certain extent by considering the analogy of these constructions with such classical results as Jacobi's identity for the theta function and the integral of the Gaussian distribution over the real line. We then discuss the ``inter-universal'' aspects of the theory, which lead naturally to the introduction of anabelian techniques.
Finally, we summarize the main abstract theoretic and diophantine consequences of inter-universal Teichm眉ller theory, which include a verification of the ABC/Szpiro conjecture. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The author considers the standard \((2n+1)\)-point finite difference approximation \(L_h\) to scalar elliptic partial differential operators \(L = \sum^n_{i = 1} c_i {\partial^2 \over \partial x^2_i}\), \(\sum^n_{i = 1} c_i = 1\), \(c_i = \text{const} > 0\), \(i = 1,2,\dots, n\), on uniform, rectangular, periodic grid domains, with the stepsize \(h\). The author provides analytic formulas for the smoothing factors \(\mu\) resulting from the point red-black Gauss-Seidel relaxation and from block relaxation strateties in the case of full and partial coarsenings.
The smoothing factors \(\mu\) are compared with the spectral radius \((\rho (M^{2h}_h))^{1/r}\) per relaxation sweep of the two-level iteration operator \(M^{2h}_h = (I_h - I^h_{2h} L^{-1}_{2h} I^{2h}_h L_h) (S_h)^r\), where \(I^h_{2h}\) and \(I^{2h}_h\) denote the usual intergrid transfer operators, and \(S_h\) is the smoothing iteration operator. The analytic formulas for \(\mu\) allow us to predict the multigrid rate as well as to adapt the smoothing procedure to the problem under consideration. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Advances in the more arithmetic branches of modern algebra and their application to number theory naturally lead, as we may venture to say today, to problems which to the well-informed mathematician either appeared familiar as part of the heritage of classical algebraic geometry or seemed to be intrinsically adapted to a solution by more conceptual geometric methods. Furthermore, since major parts of the theory of algebraic functions of one variable had been fitted into the system of algebra it was sensible that similar interpretations and attempts at solutions were (and had to be) tried for higher dimensional problems. In order to understand and appreciate the ultimate significance of this book the reader may well keep in mind the preceding twofold motivation for the interest in algebraic geometry. Classical algebraic geometry made free use of a type and mode of reasoning with which the modern mathematician often feels uncomfortable, though the experience based on a rich and intricate source of examples made the founders of this discipline avoid serious mistakes in final results which lesser men might have been prone to make.
The main purpose of this treatise is to formulate the broad principles of the intersection theory for algebraic varieties. We find those fundamental facts without which, for example, a good treatment of the theory of linear series would be difficult. The doctrine of this book is that an unassailable foundation (and thereby justification) of the basic concepts and results of algebraic geometry can be furnished by certain elementary methods of algebra. Thus, the reader will agree after some time that he is finding a delicate tool which can serve him to remove the traces of insecurity which occasionally accompany geometric reasoning. Incidentally, the term ``elementary'' used here and by the author is to be understood in a restricted technical sense, in the sense that general ideal theory and the theory of power series rings are not brought into play too often. The proofs require the general plan of using the ``principles of specialization'', as formulated algebraically by van der Waerden; and they are by no means elementary in the customary connotation. To some readers the adherence to a definite type of approach, where another author may have deemed it more instructive or appropriate to use slightly different methods, may tend to cloud occasionally immediate understanding by the less adept. However, once the reader has grasped the real geometric meaning of a definition or theorem (he then has to forget occasionally the fine points resulting from the facts that the author imposes no restriction on the characteristic of the underlying field of quantities) he will recognize how skilfully the language and methods of algebra are used to overcome certain limitations of spatial intuition.
The author begins his work with judiciously selected results from the theory of algebraic and transcendental extensions of fields [chapter I, Algebraic preliminaries]. Special emphasis has to be placed on inseparable extensions, which incidentally means a more complete account than is found in books on algebra. The further plan of the book is perhaps best appreciated if one starts to ponder over a more or less heuristic definition of ``algebraic variety'', and then asks one's self informally how one should define ``intersections with multiplicities'' of ``subvarieties''. Then, in view of the principle of local linearization in classical analysis, the author's arrangements of topics is more or less dictated by the ultimate subject under discussion, provided one does not place the
interpretation of geometrical concepts by ideal theory at the head of the discussion.
Therefore the technical definitions of point, variety, generic point and point set attached to a variety [chapter IV, The geometric language] must be preceded by suitable algebraic preparations [essentially in chapter II, Algebraic theory of specializations] and more arithmetic studies [chapter III, Analytic theory of specializations]. Crucial results in this connection, based on arithmetical considerations, are found in proposition 7 on page 60 and theorem 4 on page 62, where the existence of a well-defined multiplicity is proved for specializations. For further work, the author next introduces the concept of simple point of a variety in affine space by means of the linear variety attached to the point. [See the significant propositions 19 to 21 on pages 97--99.]
Next, the intersection theory of varieties in affine space is presented through the following stages of increasing complexity: (i) intersection with a linear subspace of complementary dimension, the 0-dimensional case, with the important criterion for multiplicity 1 in proposition 7 on page 122, and ultimately the criterion for simple points in theorem 6 on page 136; (ii) intersection with a linear subspace of arbitrary dimension, with theorem 4 on page 129 which justifies the invariant meaning of the term ``intersection multiplicity of a variety with a linear variety along a variety'' [chapter V, Intersection multiplicities, special case].
In chapter VI, entitled `General intersection theory', the results for the linear case are extended so as to culminate in the important theorem 2 on page 146 concerning the proper components of the intersection of two subvarieties in a given variety. Furthermore, all important properties of intersection multiplicities are established. Later, in appendix III, it is shown that the properties established for a certain symbol are characteristic for intersection multiplicities and uniquely define that concept. It may be mentioned that the topological definition of the chain intersections on manifolds coincides with the algebraically defined concept of this book. Of course, the underlying coefficient field has to be the field of all complex numbers and further simplifying assumptions on the variety have to be made. However, this comparison cannot be made at the level of chapters V and VI, since there one deals with affine varieties to which the ordinary
topological considerations are not directly applicable.
The subsequent chapter VII, Abstract varieties, provides the necessary background for the aforementioned connections and also contains complete proofs of those results which one might have formulated first had one deliberately adopted ideal-theoretic intentions at an early stage. The abstract varieties of this chapter are obtained by piecing together varieties in affine spaces by means of suitably restricted birational transformations. This definition of the author has turned out to be very fruitful for the work on the Riemann hypothesis for function fields and the study of Abelian varieties in general. In the course of the work, the results of the preceding chapters are extended so as to lead up to the important theorem 8 on page 193 related to Hopf's ``inverse homomorphism''. The chapter ends with a theory of cycles of dimension \(s\), that is, formal integral combinations of simple abstract subvarieties of dimension \(s\). The notion of the intersection product of cycles is also introduced here [page 202], by means of which the investigation of equivalence theories can be initiated.
This is done more explicitly in chapter IX, Comments and discussion; apparently the Riemann-Roch theorem for surfaces should now be accessible to a careful re-examination. As a further result, the theory of quasi-divisibility of Artin and van der Waerden is developed in theorems 3 and 4 on pages 224--225 and theorem 6 on page 230. These theorems exhibit the relations between the theory of cycles of highest dimension and the theory of quasi-divisibility, where naturally some of the results in appendix II, Normalization of varieties, are to be added for the necessary integral closure of the required rings of functions. In this appendix the author relates his results on the normalization of algebraic varieties to those of Zariski. At this point the individual reader may well compare the elementary and the ideal-theoretic approach to a group of theorems. In appendix I, Projective spaces, often used properties and facts concerning projective spaces are quickly developed on the basis of the preceding work. This brief discussion not only deals with results which are generally useful in algebraic geometry, but also contains one of the theorems on linear series of divisors which was frequently used in the classical work [see page 266].
Because of the wealth of material and the excellent ``advice to the reader'' prefacing this rich and important book the reviewer feels that he should mention some of the highlights and not delve into a discussion of technical details. In short, the only way to appreciate this treatise is actually to read it.
See also the review of the second ed. (1962) in Zbl 0168.18701. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. To ensure effective hull assembly line production, it is vital to consider the problems on delivery date of order and sequencing complexity within mixed-model production environments. In this paper two criteria are considered for stratified optimization according to their importance: to minimize the satisfaction ratio of delivery date, and to minimize the complexity degree of the system arising from its state frequent changes. Finding an optimal solution for this complicated problem in reasonable computational time is cumbersome. Therefore, this paper presents an improved particle swarm optimization (IPSO) algorithm to solve the multi-objective sequencing problems. Instead of modeling the positions ofparticles in a continuous value manner in traditional PSO, IPSO uses an encoding and decoding scheme of task-oriented assignment for representing the discrete input sequences of products. Furthermore, dynamic mutation operator and chaos strategy are introduced to help the particles escape from local optima and the strategy for population decomposition is proposed to further improve the efficiency of the optimization. Numerical simulation suggests that the proposed IPSO scheduler can provide obvious improvement on solution quality and running time. Finally, a case study of the optimization of a panel block assembly line was given to illustrate the effectiveness of the method. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The four-dimensional geometric commutation relation for any local operator \(\Phi\) whose classical counterpart has a definite transformation property under general coordinate transformation is proved under the concretely verifiable assumption that the equal-time geometric commutation relations are valid for all fundamental fields contained in \(\Phi\) and for their first time derivatives. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Let \(G\) be a connected graph. The eccentricity of vertex \(v\) is the maximum distance between \(v\) and other vertices of \(G\). In this paper, we study a new version of graph entropy based on eccentricity of vertices of \(G\). In continuing, we study this graph entropy for some classes of graph operations. Finally, we compute the graph entropy of two classes of fullerene graphs. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. In part XX the manifestly covariant canonical formalism of quantum gravity is extended to a coupled system of quantum gravity and a nonabelian gauge field in the Suzuki gauge, where the Suzuki gauge is a special gauge fixing in which the part corresponding to the Cartan subalgebra behaves like an abelian gauge theory. An indecomposable superalgebra is found which unifies the spacetime symmetry and the Cartan subalgebra without contradicting the no-go theorem. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Let \(q\) be a finite nonzero complex number, let the \(q\)-difference equation
\[
f(qz)f(\frac{z}{q}) = R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))} = \frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^j(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^k(z)}\tag{\(\dagger\)}
\]
admit a nonconstant meromorphic solution \(f,\) where \(\tilde{p}\) and \(\tilde{q}\) are nonnegative integers, \(a_j\) with \(0 \leq j \leq \tilde{p}\) and \(b_k\) with \(0 \leq k \leq \tilde{q}\) are polynomials in \(z\) with \(a_{\tilde{p}}\not\equiv 0\) and \(b_{\tilde{q}}\not\equiv 0\) such that \(P(z, f(z))\) and \(Q(z, f(z))\) are relatively prime polynomials in \(f(z)\) and let \(m = \tilde{p}-\tilde{q} \geq 3\). Then, (\(\dagger\)) has no transcendental meromorphic solution when \(|q| = 1\), and the lower bound of the lower order of \(f\) is obtained when \(m \geq 3\) and \(|q| \neq 1\). | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. For a review of the entire collection see [Zbl 1184.81003]. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. [For the entire collection see Zbl 0642.00029.]
The main subject of this paper is the cyclicity of extended Goppa codes over GF(q). The author uses the elements of \(PGL(2,q^ m)\) to define location sets for which the corresponding extended Goppa codes are cyclic. Furthermore he attempts to prove the following theorem. Theorem: An extended Goppa code \(\Gamma\) (L,g) where \(L=GF(q^ m)\cup \{\infty \}\) is a cyclic code if and only if there exists \(g'=(z-\beta_ 1)^ a(z-\beta_ 2)^ a\) where \(\beta_ 1\) and \(\beta_ 2\) are conjugate elements in \(GF(q^{2m})/GF(q^ m)\) and a an integer \(\geq 1\) such that \(\Gamma (L,g)=\Gamma (L,g').\)
I do not understand the second part of the authors proof where he states: ``But by construction of \(\Gamma\) (L,h), C verifies ``equation''''. My question is how the \(\beta_ i\) are determined. (Apparently independent of g, but that can not be true). Therefore according to me this Theorem is still open.
Since Theorem 4 in this paper is proven in the same way, I prefer the arguments given by \textit{Stichtenoth} (``Geometric Goppa-codes of genus 0 and their automorphism groups'' preprint). | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. For finite automata on infinite words or trees there are several natural ways to define acceptance. Compared to the cases of finite words or trees the situation is further complicated by the fact that nondeterminism properly increases the recognition power of automata. The authors have set out to clarify the relationships between six modes of accepting an infinite tree by a finite tree automaton with a set of final sets. The different definitions of acceptance are obtained by requiring that all states of any run of the automaton or, alternatively, the states infinitely often appearing in any run, form a final set, is a subset of a final set, or intersects with such a set. It is shown that for four of the acceptance modes a single final set suffices. In a series of theorems many relationships between the corresponding families of \(\omega\)-tree languages are established. However, the picture is not yet complete, and some open problems are pointed out. Finally, the authors consider the effect of the structural complexity of the recognizing automaton. In particular, they show that in all six cases the nondeterministic degree yields a proper hierarchy of language families; the nondeterministic degree of an automaton is the maximum number of choices it has in any situation. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. No review copy delivered. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. In the manifestly covariant canonical operator formalism of quantum gravity, democracy between spacetime and quantum fields (\(B\)-field, FP ghost and FP anti-ghost) is realized at the level of the superalgebra, but broken by the fact that every quantum field is a function of spacetime only. What happens if every quantum field is a function of the FP ghost? It is shown that transformation properties of fields and field equations can be nicely formulated, but trouble arises in canonical quantization. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. A pure liquid substance is inside a container. The boundary of the container is being held below the freezing point of the substance. At the beginning of the experiment the substance itself is undercooled, too. Then a crystal nucleus is added, initializing the crystallisation process. The substance freezes in tree-shaped structures called dendrites.
This doctor thesis provides a mathematical description of this physical process for the two-dimensional case. The time-depending boundary curve \(\Gamma(t)\) which separates the frozen part of the substance from the liquid part and the distribution \(\theta(x, t)\) of temperature in the course of the experiment are the main items. For a preset initial state \(\Gamma(0)\) of \(\Gamma(t)\) and given initial and boundary values of the temperature a system of differential equations for the determination of \(\Gamma(t)\) and \(\theta(x, t)\) is being worked out.
The author develops an algorithm which yields \(\Gamma(t)\) numerically as the level-curve of a piecewise polynomial FE-function. In order to obtain the temperature-function it is primarily necessary to estimate the curvature \(C_\Gamma\) of \(\Gamma(t)\) by means of numerical methods. Both, the method for the determination of \(\Gamma(t)\) and the one which yields \(\theta(x, t)\) are finally combined in one algorithm. Additionally the author presents efficient and numerically stable methods for a linear approximation of the boundary curve \(\Gamma(t)\). The paper also contains numerous examples and phenomena which may occur during dendrite growth, e.g. the possible merging of two previously separated parts. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The articles of this volume will be reviewed individually. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Using the framework provided by the mixed state representation, we numerically study the thermal behavior of a spinless bose system with a short-range repulsive interaction. The main aim of this work is to elucidate the character of the transition from a condensed phase to a non-condensed phase in a homogeneous system. Under the approximation of dilute density, the transition appears to be first order. A consistent treatment gives rise to a second-order transition. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Als Fortsetzung der ersten Mitteilung in Mem. of Coll. of Sc. Ky么to 2, 203 (F. d. M. 46, 187 (JFM 46.0187.*), 1916-18) behandelt Verf. die Zerlegbarkeitsbedingung der Ideale im eigentlichen Ring \(R\) im abstrakten Sinne. Sein Hauptresultat kann man so formulieren: Die notwendige und hinreichende Bedingung f眉r die eindeutige Zerlegbarkeit von Idealen im Ring \(R\) in Primfaktoren besteht aus: 1. f眉r jedes Maximalideal \(\mathfrak P\) ist das \(\mathfrak P^2\) enthaltende Ideal gleich \(\mathfrak P\) oder \(\mathfrak P^2,\) 2. f眉r jedes \(e\) ist \({\mathfrak P}^e \neq {\mathfrak P}^{eH}.\) Dies gilt in dem Fall, wo kein Produkt zweier von Null verschiedenen Elemente von \(R\) verschwindet. In entgegengesetztem Fall besteht die notwendige und hinreichende Bedingung nur in 1. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. See the review in Zbl 0644.46015. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. See the joint review of part I, II in [Zbl 0122.38603]. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Let \(B \in \mathbb R^n\) be a compact convex domain with a smooth boundary and containing the origin in the interior. Consider the standard lattice \(\mathbb Z^n\). It is known that the number of lattice points in the dilated domain \(tB\) equals approximately to the volume of the dilated domain, while the reminder is not greater than \(Ct^{n-1}\) asymptotically. In this paper the author proves that for almost every rotation the reminder is of order \(Ct^{n-2+2/(n+1)+\zeta_n}\) with a positive \(\zeta_n\). In the case of the plane (\(n=2\)) this estimate is further extended to general compact convex domains. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Reprint of the authors' paper in Nucl. Phys. B 405, No. 2-3, 279-304 (1993; Zbl 0908.58074). | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We obtain the equations of dual boosts (rotations in the Lorentzian dual plane \(D^2_1\)) about an arbitrary point \((H,K)\) and show that the set of all dual translations and dual boosts is a group and the set of all dual boosts is not a group. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Let \(G\) be a connected reductive algebraic group defined over a global field \(F\). Let \(\mathbb A(F)\) be the adele ring of \(F\). \(G_{\mathbb A(F)}\) is a locally compact topological group with \(G_F\) as a discrete subgroup. The group \(G_{\mathbb A(F)}\) acts on \(L^2(G_F\backslash G_{\mathbb A(F)})\). Let \(\pi\) be an irreducible representation of \(G_{\mathbb A(F)}\) which occurs in \(L^2(G_F\backslash G_{\mathbb A(F)})\). To a given \(G\) the author introduces a complex analytic group \(\hat G_F\) and to each complex analytic representation \(\sigma\) of \(\hat G_F\) and each \(\pi\) he attaches an \(L\)-function \(L(s,\sigma,\pi)\) defined by an Euler product of the local \(L\)-functions at ``unramified'' primes of \(F\). Under some natural assumptions the author proves that the Euler product converges in a half-plane.
The author's problems are mainly concerned with some fundamental properties of the \(L\)-functions:
-- Are the \(L\)-functions meromorphic in the entire complex plane with only a finite number of poles and do they satisfy the functional equation of the usual form?
-- Are there relations between the \(L\)-functions of different \(G\)?
-- Is there a relation of the \(L\)-functions to the \(L\)-functions associated to non-singular algebraic varieties (especially for \(G= \mathrm{GL}(2)\) and elliptic curves)?
The problems are posed in some reasonable precise manner. Some remarks are made about the cases where some of these problems are proved or may be proved \((G= \mathrm{GL}(1). \mathrm{GL}(2))\).
For the entire collection see [Zbl 0213.00101]. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The Davison convolution of arithmetical functions f and g is defined by
\[
(f\circ g)(n)=\sum_{d| n}f(n)g(n/d)K(n,d),
\]
where K is a complex-valued function on the set of all ordered paris (n,d) such that n is a positive integer and d is a positive divisor of n. In this paper the arithmetical equations \(f^{(r)}=g\), \(f^{(r)}=fg\), \(f\circ g=h\) in f and the congruence (f\(\circ g)(n)\equiv 0 (mod n)\), where \(f^{(r)}\) is the iterate of f with respect to the Davison convolution, have been studied. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. This paper proposes a hybrid computational method (DIPS method) which works as a simplex method for solving a standard form linear program, and, simultaneously, as an interior point method for solving its dual. The DIPS method generates a sequence of primal basic feasible solutions and a sequence of dual interior feasible solutions interdependently. Along the sequences, the duality gap decreases monotonically. As a simple method, it gives a special column selection rule satisfying an interesting geometrical property. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Bad conditioned matrix of normal equations in connection with small values of model parameters is a source of problems in parameter estimation. One solution gives the ridge estimator. Some modification of it is the aim of the paper. The behaviour of it in models with constraints is investigated as well. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Let \(X\) be an open smooth geometrically connected curve over a field \(k\subset\mathbb{C}\), and \(B_{0,n}X\) the configuration space of unordered \(n\) points on \(X\). The main purpose of this paper is to show that the \(\text{Gal} (\overline{k}/k)\)-action on the profinite fundamental group of \(B_{0,n}X\) can be completely described in terms of only the action on the profinite fundamental group of \(X\) and that of \(\mathbb{P}^1-\{0,1,\infty\}\). This is a generalization of the joint work with \textit{Y. Ihara} [On Galois actions on profinite completions of braid groups, in: Recent developments in the inverse Galois problem, Contemp. Math. 186, 173-200 (1995; Zbl 0848.11058)], which treats the case \(X=\mathbb{A}^1\).
The description tightly relates the \(\text{Gal} (\overline{k}/k)\)-actions for a positive genus curve and for \(\mathbb{P}^1- \{0,1,\infty\}\). Using this, we prove a generalization of Bely沫's injectivity theorem: for a number field \(k\), \(\text{Gal} (\overline{k}/k)\to \text{Out } \pi_1^{\text{alg}} (X\otimes_k \overline{k})\) is injective if \(X\) is an affine curve over \(k\) with non-abelian fundamental group. Also, we study field towers over \(\mathbb{Q}\) introduced by Takayuki Oda, and prove part of his conjectures. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. HyperCASE, an architectural framework for integrating CASE tools under an extended hypertext system, is described. HyperCASE's objective is to provide a powerful, user-friendly, integrated development platform that can significantly raise productivity. Its specific goal is to support software developers in project management, system analysis, design, and coding. HyperCASE integrates tools by combining a hypertext-based user interface with a common knowledge-based document repository. It includes extensive natural-language capabilities tailored to the CASE domain. These are used in the interface to the software repository, providing an alternative to hypertext information management and interdocument navigation. English input can be analyzed during informal system-requirements specification, allowing a significant degree of automation for design and concept reuse at the earliest development stages. HyperCASE's three subsystems, HyperEdit, the graphical user interface, HyperBase, the knowledge base, and HyperDict, the data dictionary, are discussed. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Verf. beweist in einem speziellen Fall die Richtigkeit der \textit{Kronecker}schen Vermutung, da脽 alle in bezug auf einen imagin盲r quadratischen K枚rper relativ \textit{Abel}schen K枚rper durch diejenigen K枚rper ersch枚pft werden, welche aus den Transformationsgleichungen der elliptischen Funktionen mit singul盲ren Moduln entstehen. Verf. beweist dies unter Benutzung der allgemeinen \textit{Hilbert}schen Ans盲tze f眉r den Fall, da脽 der Grundk枚rper der \textit{Gau脽}sche \(P{(i)}\) ist. Es ergibt sich: jeder im Bereich der rationalen komplexen Zahlen \textit{Abel}sche K枚rper ist ein Lemniskatenk枚rper. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. [For part I see Can. J. Stat. 11, 245-257 (1983; Zbl 0535.60040)].
Limit theorems for the multitype branching random walk as \(n\to \infty\) are given (n is the generation number) in the case in which the branching process has a mean matrix which is not positive regular. In particular, the existence of steady state distributions is proven in the subcritical case with immigration, and in the critical case with initial Poisson random fields of particles. In the supercritical case, analogues of the limit theorems of \textit{H. Kesten} and \textit{B. P. Stigum} [Ann. Math. Stat. 37, 1211-1223 and 1463-1481 (1966; Zbl 0203.174)] are given. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. [For the entire collection see Zbl 0741.00019.]
Recent studies on ``some new relations among the classical objects of the title'' developed by the author together with G. Anderson, R. Coleman, P. Deligne, V. G. Drinfeld and other authors are surveyed. The contents are based on the author's lecture at the Plenary Sessions of ICM90 at Kyoto.
Let \(X_ n\) (\(n=4,5,\dots\)) be the configuration space defined as the \(n\)-times product of \(\mathbb{P}^ 1\) minus the weak diagonals modulo \(PGL(2)\)-action. Then the exterior actions of the Galois group \(G_ \mathbb{Q}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on the ``braid groups'' \(\widehat \pi_ 1(X_ n)\) (\(n\geq 4\)) arise, which we denote \(\varphi_ n: G_ \mathbb{Q}\to\text{Aut }\widehat\pi_ 1(X_ n)\). Some compatibilities of \(\varphi_ 4\) and \(\varphi_ 5\) enable one to obtain a homomorphism of \(G_ \mathbb{Q}\) into the Grothendieck-Teichm眉ller group \(\widehat G T\) which is a certain specified subgroup of \(\text{Aut }\widehat \pi_ 1(X_ 4)\) introduced by Drinfeld in the context of a certain transformation group of the quasi-triangular quasi-Hopf algebra structures. When \(\widehat \pi_ 1(X_ n)\) are replaced by their pronilpotent quotients, it is possible to study the representations \(\varphi_ n\) in more detail. There is an elementwise explicit description of \(\varphi_ 4(\sigma)\) ``modulo double commutator subgroup of \(\pi_ 1\)'' for \(\sigma\in G_ \mathbb{Q}\) due to Anderson/Coleman/Ihara- Kaneko-Yukinari, which also gives a universal expression of Jacobi sums as the ``adelic beta functions''. The determination of the Galois image of \(\varphi_ n\) is basically still open, but there are elaborate approaches independently by the author and P. Deligne by linearizing the situation via lower central series of \(\widehat \pi_ 1\). Several fundamental questions in the field are raised concretely along the contexts. Among them, Question 5.3.4(i) is now solved affirmatively by the author [``On the stable derivation algebra associated with some braid groups.'' Isr. J. Math. (to appear)]. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Inhalt des auf Vorlesungen basierenden Buches: 1. Grundlagen (stark motivierende Fragen: Ist ``alles'' berechenbar ? Beweise f眉r die Undurchf眉hrbarkeit gewisser Berechnungen). 2. Programme (Berechenbarkeit; verschiedene Programmiersprachen und ihre 脛quivalenz; Halteproblem in PASCAL als (mitrei脽end knapp, streng, einleuchtend durchgef眉hrter) Beweis f眉r Nichtberechenbarkeit einer Funktion. 3. Funktionen (begeisternd formalisiert vorgetragenes klassisches Wissen 眉ber arithmetische Funktionen). 4. Regelsprachen. 5. Regul盲re Sprachen und Automaten. 6. Kontextfreie Sprachen (klassisches Grundwissen 眉ber Regelsprachen - bis auf einige Ausnahmen im holpernden Duktus bekannter Literatur; der Ansatz: ``Produktionssysteme und Automaten definieren Relationen 眉ber Wortmengen'' bleibt Abbreviatur statt zum Formalismus der Relationen ausgebaut zu werden.) 7. Berechenbarkeit (脺bersichtliche Einf眉hrung der Turingmaschinen; der Beweis ihrer 脛quivalenz zu MiniPascal-Programmen ist die wesentliche didaktische Leistung des Buches - so werden bereits bewiesene Resultate nutzbar und der bekannte ``Turing-Wust'' wird vermieden. Durch geschickte Reduktion auf bereits Bewiesenes k枚nnen das PKP und die anderen klassischen Entscheidbarkeitsfragen auf wenigen Seiten abgehandelt werden.) 8. Komplexit盲t (guter, in seinen arithmetischen Betrachtung wiederum begeisternder 脺berblick 眉ber Zeitschranken, Komplexit盲tsmasse und - klassen. Reduzibilit盲t und Vollst盲ndigkeit werden auf kleinstem Raum anschaulich abgehandelt.) Durch dies Buch weist sich Verf. vor allem als passionierter Mathematiker mit Vorlieben im Bereich des klassisch- arithmetischen Kalk眉ls aus. Dort gelingt ihm jene Eleganz der Beweisf眉hrung, die den Leser seine Umgebung vergessen l盲脽t. Ohne es explizit zu intendieren - sozusagen als Seiteneffekt - f眉hrt er au脽erdem den beweisf盲higen Umgang mit der ``PEANO-Datenstruktur'' vor. Die Unvollkommenheiten des klassischen Einstiegs in die Theorie der Regelsprachen erinnern zwar schmerzhaft an das Vorlesungsmanuskript als Urbild dieses Buches, k枚nnen jedoch als Anreiz verstanden werden, den Relationskalk眉l 眉ber Halbgruppen in Richtung jener Herleitungsf盲higkeit auszubauen, die ohne ``Prosa'' auskommt. Kurzum: Eine in jeder Hinsicht anregende Lekt眉re. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The M-decomposition is defined for a graph, representing a large-scale system of nonlinear equations, with specified entrance and exit vertices, in terms of the Menger-type linkings from the entrance to the exit. Some properties of the M-decomposition are shown; in particular it is noted that the M-decomposition agrees with the Dulmage-Mendelsohn decomposition of the associated bipartite graph. The M-decomposition is useful for the structural analysis of systems of equations - it leads to the finest block-triangularization and the resulting subproblems are structurally solvable. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We prove the existence of new-type solutions of the modified Fischer-Kolmogorov equation with slow/fast diffusion and with possibly nonsmooth double-well potential. We show that a certain relation between the rate of the diffusion and the smoothness of the potential may originate new type solutions which do not occur in the classical Fischer-Kolmogorov equation. The main focus of this paper is to show the sensitivity of the mathematical modelling with respect to the chosen form of the diffusion term and the shape of the double-well potential. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We point out that the elliptic genus of the \(K3\) surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group \(M_{24}\). The reason remains a mystery. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The discrete parallel machine makespan scheduling location (ScheLoc) problem is an integrated combinatorial optimization problem that combines facility location and job scheduling. The problem consists in choosing the locations of multiple machines among a finite set of candidates and scheduling a set of jobs on these machines, aiming to minimize the makespan. Depending on the machine location, the jobs may have different release dates, and thus the location decisions have a direct impact on the scheduling decisions. To solve the problem, it is proposed a new arc-flow formulation, a column generation and three heuristic procedures that are evaluated through extensive computational experiments. By embedding the proposed procedures into a framework algorithm, we are able to find proven optimal solutions for all benchmark instances from the related literature and to obtain small percentage gaps for a new set of challenging instances. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We prove potential modularity theorems for \(l\)-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral \(j\)-invariant defined over a totally real field. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. In flux compactifications of M-theory a superpotential is generated whose explicit form depends on the structure group of the seven-dimensional internal manifold. In this paper, we discuss superpotentials for the structure groups: \(G_2\), \(SU(3)\) or \(SU(2)\). For the \(G_2\) case all internal fluxes have to vanish. For \(SU(3)\) structures, the non-zero flux components entering the superpotential describe an effective one-dimensional model and a Chen-Simons model if there are \(SU(2)\) structures. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. [For the entire collection see Zbl 0549.00011.]
Betrachtet wird eine instation盲re, inkompressible, nichtviskose und rotationsfreie Str枚mung in einem Ringgebiet \(1\leq r\leq \gamma (t,\theta)\) die Ebene (0\(\leq \theta \leq 2\pi)\), wobei der 盲u脽ere ''freie'' Rand durch eine Gleichgewichtsbedingung und durch Vorgabe der Gesamtfl盲che des Ringgebiets festgelegt wird. Im station盲ren Fall gibt es eine triviale L枚sung mit \(\gamma (t,\theta)=r_ 0\). Der Hauptsatz weist f眉r gen眉gend kleine Abweichung des Anfangsgebietes vom Ring \(1\leq r\leq r_ 0\), d.h. bei hinreichend kleinem \(\| \gamma (0,.)-r_ 0\|\) in geeigneter Norm die Existenz einer eindeutigen L枚sung sowohl hinsichtlich des Randes \(\gamma\) (t,\(\theta)\) als auch hinsichtlich der Stromfunktion V(t,r,\(\theta)\) nach.
Dies ist ein recht kompliziertes Unterfangen: Man startet mit \(\gamma_ u(t,\theta)=\gamma (0,\theta)+u(t,\theta),\) sorgt f眉r die richtige Fl盲che des Rings \(1\leq r\leq \gamma_ u(t,\theta)\), l枚st in diesem Ring die Potentialgleichung (mit festem t) \(\nabla V=0\) mit \(V=0\) auf \(r=1\) und
\[
V=\int^{\theta}_{0}\gamma_ u(t,\phi)(\partial \gamma_ u/\partial t)(t,\phi)d\phi \quad auf\quad r=\gamma_ u(t,\theta),
\]
geht mit der (von u abh盲ngigen L枚sung) \(V=V_ u\) in die Cauchy-Riemannsche Gleichungen
\[
(1/r)\partial^ 2V_ u/\partial t\partial \theta +\partial q_ u/\partial r=0,\quad -\partial^ 2V_ u/\partial t\partial r+(1/r)\partial q_ u/\partial \theta =0,
\]
wodurch \(q_ u\) im wesentlichen festgelegt wird (diese Gleichungen entstammen den Euler-Gleichungen) und mu脽 schlie脽lich noch die Randbedingung am freien Rand erf眉llen:
\[
F(\gamma (0,.),u):=(\partial /\partial \theta)([q_ u-1/2\| \nabla V_ u\|^ 2+g/r]|_{r=\gamma_ u(\theta,t\quad)}-\sigma \kappa_{u(t)})=0
\]
\((\kappa_ u\) ist dabei die Kr眉mmung von \(r=\gamma_ u(t,\theta))\). Diese nichtlineare Gleichung f眉r u hat nun eine eindeutige L枚sung, wenn \(\gamma\) (0,.) nicht zu weit von \(r_ 0\) abweicht - dies folgt (genauer: soll folgen, denn der Autor verweist hinsichtlich der sicher sehr aufwendigen Details auf ein unspezifiziertes ''anderswo'') aus einem verallgemeinerten Satz 眉ber implizite Funktionen. Mit einigen Bemerkungen zur Stabilit盲t bestimmter station盲rer L枚sungen schlie脽t die Arbeit ab. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We consider likelihood and Bayes analyses for the symmetric matrix von Mises-Fisher (matrix Fisher) distribution, which is a common model for three-dimensional orientations (represented by \(3\times 3\) orthogonal matrices with a positive determinant). One important characteristic of this model is a \(3\times 3\) rotation matrix representing the modal rotation, and an important challenge is to establish accurate confidence regions for it with an interpretable geometry for practical implementation. While we provide some extensions of one-sample likelihood theory (e.g., Euler angle parametrizations of modal rotation), our main contribution is the development of MCMC-based Bayes inference through non-informative priors. In one-sample problems, the Bayes methods allow the construction of inference regions with transparent geometry and accurate frequentist coverages in a way that standard likelihood inference cannot. Simulation is used to evaluate the performance of Bayes and likelihood inference regions. Furthermore, we illustrate how the Bayes framework extends inference from one-sample problems to more complicated one-way random effects models based on the symmetric matrix Fisher model in a computationally straightforward manner. The inference methods are then applied to a human kinematics example for illustration. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. J. Leray considered a backward self-similar solution of the Navier-Stokes equations in the hope that it gives us an example of the finite-time blow-up of the three dimensional nonstationary Navier-Stokes equations. However, he showed no example of solutions. We list here some particular solutions and discuss their fluid mechanical properties. We also consider a forward self-similar solution, which describes solutions decaying as time tends to infinity. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. For a given group \(G\), let \(G^{mab}\) denote the meta-abelianization of \(G\); i.e., \(G^{mab} =G/G''\) where \(G''\) is the second commutator subgroup of \(G\). For a commutative ring \(A\), let \(\text{St} (2,A)\) be the Steinberg group of rank one of \(A\) and let \(K_2(2,A)\) be the kernel of the natural homomorphism \(\pi:\text{St} (2,A) \to\text{SL} (2,A)\).
The following results are proved:
Theorem 1. Let \(p\) be a prime number. Then
\[
\text{SL}\bigl(2, \mathbb{Z}[1/p] \bigr)^{mab} \simeq \begin{cases} \mathbb{Z}_3 \ltimes (\mathbb{Z}_2 \times \mathbb{Z}_2) \quad & p=2 \\ \mathbb{Z}_4 \ltimes\mathbb{Z}_3 \quad & p=3 \\ \mathbb{Z}_{12} \ltimes (\mathbb{Z}_2 \times \mathbb{Z}_6) \quad & p\geq 5 \end{cases}.
\]
Theorem 2.
1. Suppose \(p=2,3\). Then \(K_2(2, \mathbb{Z}[1/p]) \simeq \mathbb{Z} \times \mathbb{Z}_{p-1}\), and \(K_2(2,\mathbb{Z} [1/p])\) is central in \(\text{St} (2,\mathbb{Z} [1/p] )\).
2. Suppose \(p\geq 5\). Then \(K_2(2, \mathbb{Z}[1/p]) \supset \mathbb{Z}\times \mathbb{Z}\), and \(K_2(2, \mathbb{Z} [1/p])\) is not central.
The meta-abelianization of \(\text{SL} (2,\mathbb{Z} [1/p])\) is computed from that of \(\text{St} (2,\mathbb{Z} [1/p])\) together with finding appropriate elements of \(K_2(2, \mathbb{Z} [1/p])\) described by Dennis-Stein symbols. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. [For the entire collection see Zbl 0672.00003.]
This is a set of three very inspiring lectures on the connection between the KP hierarchy and infinite dimensional Grassmann manifolds. The author describes the universal Grassmannian, the three approaches to the complete integrability of the KP hierarchy and how the approach based on conjugation by a micro-differential operator of order zero recasts the KP hierarchy as a dynamical system on the universal Grassmannian. Then the KP hierarchy is formulated in terms of deformations of modules over the differential operators. Finally, a very provocative multidimensional generalization is proposed and its link with algebraic geometry is explored. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. In an optimal control problem for quasi linear elliptic equations the well-posedness matters are studied and necessary conditions of optimality are established. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The expression for the equal-time commutator between \(b_{\rho}\) and \(\dot b_{\rho}\), which was derived previously from the consistency requirement of the BRS transformation, is proved on the basis of canonical commutation relations and field equations. It is found that the commutators involving \(b_{\rho}\) exhibit `tensor-like' behavior quite analogously to general covariance.
The manifestly covariant quantum field theory is formulated for the coupled Einstein-Maxwell system. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Inspired by a question of P. A. Smith various authors constructed one fixed point actions on homotopy spheres. To get a deeper understanding, others determined the smallest dimension in which such an action might occur. There are answers for specific groups and in general. The paper under review is part of an effort, for specific groups, to list dimensions in which there are no actions on integral homology spheres with an odd number of fixed points. At times coefficients different from \(\mathbb Z\) are used.
For the groups \(A_6\) and \(\mathrm{SL}(2,9)\) the author lists dimensions in which there is no action on a \(\mathbb Z_2\) homology sphere with an odd number of fixed points. For \(S_6\), \(\mathrm{PGL}(2,9)\), the Mathieu group \(M_{10}\) and \(\mathrm{Aut}(A_6)\) the author lists dimensions in which there is no action on an integral homology sphere with an odd number of fixed points. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We consider the problem of finding a maximum common subbase of two submodular systems on \(E\) with \(| E|=n\). First, we present a new algorithm by finding the shortest augmenting paths, which begins with a pair of subbases of the given submodular systems and is convenient for adopting the preflow-push approach of Goldberg and Tarjan. Secondly, by using the basic ideas of the preflow-push method, we devise a faster algorithm for the intersection problem, which requires \(O(n^ 3)\) push and \(O(n^ 2)\) relabeling operations in total by the largest-label implementation with a specific order on the arc list of each vertex in the auxiliary graph. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Proximal SVM (PSVM), which is a variation of standard SVM, leads to an extremely faster and simpler algorithm for generating a linear or nonlinear classifier than classical SVM. An efficient incremental method for linear PSVM classifier has been introduced, but it can't apply to nonlinear PSVM and incremental technique is the base of online learning and large data set training. In this paper we focus on the online learning problem. We develop an incremental learning method for a new nonlinear PSVM classifier, utilizing which we can realize online learning of nonlinear PSVM classifier efficiently. Mathematical analysis and experimental results indicate that these methods can reduce computation time greatly while still hold similar accuracy. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. A method is introduced for determining the critical initial imperfection of discretized structures that decreases the load-bearing capacity most rapidly. The effects of imperfections on simple critical points, such as limit points of loads and simple bifurcation points, are theoretically investigated based on the idea of the Lyapunov-Schmidt decomposition developed in bifurcation theory. Imperfection sensitivity varies with the types of points. Nonetheless critical imperfection pattern is expressed in the same formula regardless of the types. Among various imperfections, the most influential can be found in a quantitative manner. The validity and the usability of the poposed method are illustrated through its application to simple example structures. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and present the study of its numerical properties. By following [\textit{A. Buffa} and \textit{C. Giannelli}, Math. Models Methods Appl. Sci. 26, No. 1, 1--25 (2016; Zbl 1336.65181); Math. Models Methods Appl. Sci. 27, No. 14, 2781--2802 (2017; Zbl 1376.41004); \textit{A. Buffa} et al., Comput. Aided Geom. Des. 47, 83--92 (2016; Zbl 1418.65011)], optimal convergence rates of the AIGM can be proved when suitable approximation classes are considered. This is in line with the theory of adaptive methods developed for finite elements, recently well reviewed in [\textit{R. H. Nochetto} and \textit{A. Veeser}, Lect. Notes Math. 2040, 125--225 (2012; Zbl 1252.65192)]. The important output of our analysis is the definition of classes of admissibility for meshes underlying hierarchical splines and the design of an optimal adaptive strategy based on these classes of meshes. The adaptivity analysis is validated on a selection of numerical tests. We also compare the results obtained with suitably graded meshes related to different classes of admissibility for 2D and 3D configurations. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Es handelt sich um den unver盲nderten Wiederdruck der bekannten und gesch盲tzten Seminar Notes des Artin-Tate-Seminars an der Universit盲t von Princeton 1951/52 (herausgegeben: Harvard, 1961). Die Verff. behandeln darin -- unter Verwendung des Idealbegriffs und kohomologischer Methoden -- die Klassenk枚rpertheorie der globalen K枚rper, d.h. der algebraischen Zahlk枚rper und der Funktionenk枚rper einer Variablen 眉ber endlichem Konstantenk枚rper. Die fehlenden ersten vier Kapitel, die Kohomologietheorie und lokale Klassenk枚rpertheorie beinhalten und zum Verst盲ndnis des Buches erforderlich sind, werden in der Einleitung skizziert. Zu einer gr眉ndlichen Erarbeitung dieser Theorien seien dem Leser das Buch von \textit{J. Neukirch} [Klassenk枚rpertheorie. Bonn. Math. Schr. 26, 296 p. (1967; Zbl 0165.36602)] oder auch die Lecture Notes einer von Artin 1950/51 in Princeton gehaltenen Vorlesung [\textit{E. Artin}, Algebraic numbers and algebraic functions. New York etc.: Gordon and Breach (1967; Zbl 0194.35301)] empfohlen.
Inhalts眉bersicht: Einleitung. Kapitel 5: Erste fundamentale Ungleichung. Kapitel 6: Zweite fundamentale Ungleichung. Kapitel 7: Reziprozit盲tsgesetz. Kapitel 8: Existenzsatz. Kapitel~9: Zusammenhangskomponente der Idelklassen. Kapitel 10: Satz von Grunwald. Kapitel~11: Theorie der h枚heren Verzweigungen. Kapitel 12: Explizite Reziprozit盲tsgesetze. Kapitel 13: Gruppenerweiterungen. Kapitel 14: Abstrakte Klassenk枚rpertheorie. Kapitel 15: Weilsche Gruppen.
Zur Inhalts眉bersicht einige erg盲nzende Bemerkungen: Die Kapitel 5--8 enthalten die Beweise der grundlegenden S盲tze der globalen Klassenk枚rpertheorie; eine ausf眉hrliche Theorie der Klassenformation befindet sich in Kapitel 14. Ausgehend von einer lokal-global Beziehung f眉r \(m\)-te Potenzen wird in Kapitel 9 die Struktur der Zusammenhangskomponente der Eins in der Idel\-klassengruppe bestimmt und in Kapitel 10 u.a. der auf \textit{W. Grunwald} [J. Reine Angew. Math. 169, 103--107 (1933; Zbl 0006.25204)] zur眉ckgehende Satz bewiesen, dass es zu gegebenem glo\-balen K枚rper \(k\), gegebener endlicher Menge von Primdivisoren \(\mathfrak p\) in \(k\) und gegebenen m枚glichen Erweiterungsgraden \(n_{\mathfrak p}\) von \(k_{\mathfrak p}\) stets eine zyklische Erweiterung \(K/k\) gibt, deren Grad das klein\-ste gemeinsame Vielfache der \(n_{\mathfrak p}\) ist und f眉r die die lokalen Erweiterungen \(K_{\mathfrak p}/k_{\mathfrak p}\) vom Grade \(n_{\mathfrak p}\) sind.
Kapitel 11 enth盲lt neben der Theorie der h枚heren Verzweigungen eine ``allgemeine lokale Klassen\-k枚rpertheorie'' (d.h. die zugrunde liegenden vollst盲ndigen K枚rper sind allgemeiner diejenigen, die einen vollkommenen Restklassenk枚rper haben, f眉r den es zu jedem nat眉rlichen \(n\) genau eine Erweiterung \(n\)-ten Grades gibt.
In Kapitel 12 wird mit Hilfe der lokalen Analysis das Normenrestsymbol in gewissen Kummerschen K枚rpern explizit bestimmt und ein Reziprozit盲tsgesetz f眉r das \(m\)-te Potenzrestsymbol bewiesen, das das quadratische Reziprozit盲tsgesetz als Spezialfall enth盲lt.
Kapitel 13 enth盲lt u.a. einen sch枚nen, weitgehend kohomologischen Beweis des Hauptidealsatzes und Kapitel 15 die L枚sung (Shafarevich--Weil) der von Hasse gestellten Aufgabe, f眉r eine Galoissche K枚rpererweiterung \(L\supset K\supset k\), \(L/K\) abelsch, die Invarianten der zugeh枚rigen Gruppen\-erweiterung innerhalb \(K\) zu charakterisieren.
Auf folgende ausgezeichnete B眉cher 眉ber Klassenk枚rpertheorie sei hingewiesen: \textit{J. W. S. Cassels} und \textit{A. Fr枚hlich}, Algebraic number theory, London, New York: Academic Press (1967; Zbl 0153.07403) und \textit{A. Weil}, Basic number theory. Grundlehren Math. Wiss. Band 144. Berlin etc.: Springer (1967; Zbl 0176.33601). | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. Ein B眉chelchen, welches wohl nichts Neues, aber manche schweren Fehler enth盲lt. | 0 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The goal of the article under review is to extend the methods of \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 443--551 (1995; Zbl 0823.11029)] and \textit{R. Taylor, A. Wiles} [Ann. Math. (2) 141, No. 3, 553--572 (1995; Zbl 0823.11030)] from \(\mathrm{GL}_2\) to unitary groups of any rank.
The authors work with the disconnected group \(\mathcal{G}_n\), which is the semidirect product of \(\mathrm{GL}_n\times \mathrm{GL}_1\) by the two element group. In this setting the Taylor-Wiles argument carries over well, and the authors prove \(R=T\) type of theorems in the ``minimal case'' (i.e., they consider deformation problems where the lifts on the inertia groups away from \(\ell\) are completely prescribed, e.g., as unramified as possible away from \(\ell\)). As the authors indicate in their very well-written introduction, there are three key inputs to their proof:
(1) \textit{F. Diamond} [Invent. Math. 128, No. 2, 379--391 (1997; Zbl 0916.11037)] and K. Fujiwara's observation that Mazur's multiplicity one principle is not needed for the Taylor-Wiles argument.
(2) A trick due to \textit{C. Skinner} and \textit{A. Wiles} [Duke Math. J. 107, No. 1, 15--25 (2001; Zbl 1016.11017)] which involves a base-change argument to bypass Ribet's level lowering results.
(3) The proof of the local Langlands conjecture for \(\mathrm{GL}_n\) by the second and third named author, and its compatibility with the global Langlands correspondence.
The authors prove the automorphy of certain Galois representations \(r: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GSp}_n(\mathbb{Z}_\ell)\) under a list of technical conditions. We do not state these hypotheses in detail, instead we give an overview of their content below. We refer the reader to Theorems 4.4.2 -- 4.4.3 and Corollary 4.4.4 of the paper under review.
(\(H_I\)) The image of the representation \(\overline{r}:= r \bmod {\ell}\) restricted to \(\text{Gal}(\overline{F}/F(\zeta_\ell))\) should be big in the sense of Definition 2.5.1 of the article, so that the Chebotarev argument in the Taylor-Wiles method works. This condition holds, for example, when the image of \(\overline{r}\) contains \(\mathrm{Sp}_n(\mathbb{F}_\ell)\) (see Lemma 2.5.5).
(\(H_{C+HT}\)) We have \(\ell>n\) and \(r \big{|}_{\text{Gal}(\overline{\mathbb{Q}}_\ell/\mathbb{Q}_\ell)}\) is crystalline with Hodge-Tate weights lying within the interval \([0,n-1]\). This is to ensure that Fontaine-Laffaille theory applies to calculate the local deformation ring at \(\ell\). Furthermore, the Hodge-Tate weights are assumed to be distinct. This, together with the assumption that \(r\) is valued in the symplectic group ensures that \(r\) satisfies the sort of self-duality which is needed for the numerical criterion in Taylor-Wiles method to hold.
(\(H_S\)) For a non-empty auxiliary set of primes \(S\) and for \(\ell \neq q \in S\), the restriction \(r\big{|}_{G_{\mathbb{Q}_q}}\) ``looks as if it could correspond to a Steinberg representation under the local Langlands correspondence''. The authors explain that the set \(S\) is required to be non-empty so that relevant automorphic forms may be transferred to and from unitary groups, so that one can attach Galois representations to these automorphic forms.
(\(H_M\)) Outside the auxiliary set \(S\) and away from \(\ell\). the image of \(r\) on the inertia should be finite. This hypothesis has to do with the fact that the authors are working in the minimal case.
In order to be able to remove the minimality condition (\(H_M\)), the authors formulate a conjecture (Conjecture I and the stronger Conjecture II, both in Section 5.3), which is an analogue of Ihara's Lemma for elliptic modular forms ([\textit{Y. Ihara}, Discrete Subgroups of Lie Groups Appl. Moduli, Pap. Bombay Colloq. 1973, 161--202 (1975; Zbl 0343.14007)], \textit{K. Ribet} [Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 503--514 (1984; Zbl 0575.10024)]). If Conjecture I holds, the authors obtain certain level raising results which, much like in [ A. Wiles, loc. cit.], can be used to treat the non-minimal case.
The third named author has developed a new technique in a sequel to this paper [\textit{R. Taylor}, Publ. Math., Inst. Hautes 脡tud. Sci. 108, 183--239 (2008; Zbl 1169.11021)], which may be used to treat the non-minimal case without the relying on Conjecture I. The authors indicate that the results of the paper under review are still stronger than that of R. Taylor (loc. cit.) (assuming Conjecture I), as in loc.cit., a Hecke algebra is identified with a universal deformation ring modulo its nilradical.
Overall, this is a very important paper which is written very nicely; it contains very many details on the most recent technology involved in the proofs of modularity lifting theorems. | 1 |
This is the unique book about the Reseach Institute for Mathematical Sciences (RIMS) at Kyoto University, consisting of 8 chapters from Chapter 0 to Chapter $\infty$.
Chapter 0 is abundant in interesting episodes on how RIMS was finally established.
Chapter 1, consisting of three sections together with one column, is concerned with so-called \textit{Sato school}. \S 1 addresses the birth of \textit{algebraic analysis}, characters on the scenes being Mikio Sato, Hikosaburo Komatsu, Kosaku Yosida, Mitsuo Morimoto, Masaki Kashiwara, Takahiro Kawai.
\S 2 explains how the theory of \textit{Sato hyperfunctions} emerged.
The column inserted between \S 2 and \S 3 discusses the relationship between Sato hyperfunctions and numerical analysis, speaking in terms of characters, the interaction between Mikio Sato, Hidetoshi Takahashi (a leading figure in the production of the prototype parametron-based computers PC-1 in Japan in the 1950s) and Masatake Mori [Zbl 0225.65026; Zbl 0267.65016].
\S 3 deals with mathematical physics of Sato school, which was driven energetically by the second generation of Sato school. The major figures in the first generation of Sato school are Masaki Kashiwara and Takahiro Kawai, whose achievement has culminated in [Zbl 0277.46039]. The major figures in the second generation of Sato school are Michio Jimbo and Tetsuji Miwa, the former having become a graduate student under the direction of Sato at RIMS in 1974 and the latter having become an assistant professor at RIMS in 1973. The first work of Sato school in mathematical physics is concerned with so-called Ising model, which has resulted, in 1977, in what is called \textit{holonomic quantum fields}. Sato believed firmly that there should be some beautiful mathematical structure behind any class of exactly soluble differential equations. The second work of Sato school in mathematical physics is concerned with soliton equations, which turned out to be no other than Pl眉cker relations and whose solutions turned out to form an infinite-dimensional Grassmannian manifolds [Zbl 0688.58016; Zbl 0528.58020; Zbl 0507.58029].
The chapter is concluded with a chronololgy of Sato school.
Chapter 2, consisting of two sections and two columns, is concerned with \textit{algebraic geometry}. \S 1 depicts Heisuke Hironaka and his work on \textit{resolutions on singularities of algebraic varieties} [Zbl 1420.14031; Zbl 0122.38603], characters on the scene being Yasuo Akizuki, David Mumford, Michael Artin, Oscar Zariski and Alexander Grothendieck.
\S 2 addresses Shigefumi Mori and his work on the \textit{classification of algebraic three-folds} [Zbl 0773.14004; Zbl 1103.14301; Zbl 0926.14003; Zbl 1001.00023], characters on the scene being Koji Doi, Masayoshi Nagata, Masaki Maruyama, Hideyasu Sumihiro, Miles Reid, Teruhisa Matsusaka and J谩nos Koll谩r. The first column is an introduction to algebraic geometry. The second column, concerned with algebraic geometry at Kyoto University, begins with
Masazo Sono (1886--1969) [JFM 46.0187.01; JFM 47.0891.04]. After the second war,
Yasuo Akizuki [Zbl 0005.38701; Zbl 0005.00701; Zbl 0012.24502; Zbl 0012.24501], having become a professor at Kyoto University in 1948, established a platform of collecting mathematical talent in Kyoto.
One of the collected talent was
Jun-ichi Igusa [Zbl 0045.15803; Zbl 0045.32501; Zbl 0045.32502; Zbl 0054.06404] who served Kyoto University from 1949 to 1953 when he left for Zariski in Harvard. Akizuki learned Hodge theory from Igusa to write a book on it. The Akizuki school in the 1950s included
Masayoshi Nagata [Zbl 0045.00603; Zbl 0039.26303; Zbl 0042.02901; Zbl 0045.16003; Zbl 0054.01802],
Kotaro Okugawa [Zbl 0053.02001; Zbl 0056.26404],
Shigeo Nakano [Zbl 0058.17202; Zbl 0059.14701; Zbl 0068.34403; Zbl 0068.34501],
Yoshikazu Nakai [Zbl 0079.36901; Zbl 0079.36803; Zbl 0066.14602; Zbl 0077.34304],
Teruhisa Matsusaka [Zbl 0045.42101; Zbl 0045.24201; Zbl 0045.42102],
Yoshiro Mori [Zbl 0068.26304; Zbl 0053.21701],
Mieo Nishi [Zbl 0064.26508; Zbl 0068.14801; Zbl 0068.34504] and
Hideyuki Matsumura [Zbl 0055.26606; Zbl 0079.36802; Zbl 0079.36804].
Hironaka studied algebraic geometry, as a college student or a graduate student, under the direction of Yasuo Akizuki. When Shigefumi Mori was in the second year of college, Professor Doi advised him to read Weil's trilogy on algebraic geometry [Zbl 0063.08198, Zbl 0037.16202, Zbl 0036.16001], which he had read through for half a year. Since Doi became too busy to take care of Mori, Mori continued to study algebraic geometry under the direction of Nagata, settling a question posed by Nagata to write a paper ``On automorphisms of affine planes'' as a 4th year university student, which was published in a K么ky没roku of RIMS.
The chapter is concluded with chronologies of Heisuke Hironaka and Shigefumi Mori.
Chapter 3, consisting of three sections together with one column, is concerned with \textit{number theory}. RIMS started without any number theorist, and it was in 1989, when RIMS had already passed 26 years since its foundation, and the incumbent director of RIMS at that time was Mikio Sato, who is versed not only in algebraic analysis but also in number theory, that a number theorist called Yasutaka Ihara arrived at RIMS from the University of Tokyo.
\S 1 addresses the classics of Japan's number theory, namely, Teiji Takagi and his class field theory [JFM 47.0147.03; JFM 48.0169.01; Zbl 0176.33504]. Class field theory is at the bottom of Japan's number theory, and the University of Tokyo was its stronghold whose first generation was Masao Sugawara [Zbl 0013.19602; Zbl 0013.38902], Shokichi Iyanaga [JFM 55.0103.07] and Yukiyoshi Kawada [Zbl 0019.24704], and whose second generation was Goro Shimura [Zbl 0142.05402; Zbl 0141.38704], Yutaka Taniyama [Zbl 0213.22803; Zbl 0090.25703] and Kenkichi Iwasawa [Zbl 0090.02903; Zbl 0093.04403; Zbl 0202.33102]. In the international symposium on algebraic number theory [Zbl 0071.26501], Andr茅 Weil met Taniyama and Shimura. All the three arrived almost simultaneously at the idea of algebra-geometric treatment of \textit{complex multiplication}.
\S 2 is concerned with Yasutaka Ihara and his \textit{nonabelian class field theory} [Zbl 0231.12017], which has greatly influenced the development of the so-called Langlands program [Zbl 0225.14022; Zbl 0225.14023] and Pierre Deligne in France, and which is indeed a rainbow bridge between the Riemann zeta function and the Selberg zeta function. In the 1980s Ihara was interested in action of Galois groups on fundamental groups [Zbl 0757.20007]. After the arrival of Ihara at RIMS, Takayuki Oda [Zbl 0718.11021; Zbl 0958.11037; Zbl 0812.11033] and Makoto Matsumoto [Zbl 0900.14001; Zbl 0867.14011; Zbl 0858.12002] joined the number-theoretic group at RIMS, which was followed by Akio Tamagawa [Zbl 1206.11081; Zbl 1194.14044], Takeshi Tsuji [Zbl 1423.14147; Zbl 1342.14045] and Shinichi Mochizuki. The Grothendieck conjecture on anabelian geometry was partially solved by Hiroaki Nakamura and Akio Tamagawa, and was finally solved completely by Shinichi Mochizuki [Zbl 0943.14014] exploiting his $p$-adic Hodge theory [Zbl 1091.14501]. Ihara, who prefers the delicate music of Wolfgang Amadeus Mozart, depicted the work of the three as Wagner-like, saying that the framework is grandiose while \textit{anabelian geometry} seems to beat the air. In March 2002 Ihara retired from RIMS.
\S 3 is concerned with Mochizuki's \textit{inter-universal Teichm眉ller theory} [Zbl 1403.14061] and his alleged solution on the \textit{$abc$ conjecture}. Here I refrain from doing more than only giving a sincere sincere report of Peter Scholze and Jacob Stix on Mochizuki's work [\url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf} (opens in new tab)] and an article in Nature [\url{https://www.nature.com/articles/d41586-020-00998-2}]. Then comes a column on number theory of Mikio Sato. In 1960s Sato was actively engaged in number theory. It was the Ramanujan conjecture that Sato first tackled in number theory. This led Sato to the so-called \textit{Sato-Tate conjecture} [Zbl 0213.22804], which was settled by Richard Taylor and others [Zbl 1169.11021; Zbl 1169.11020; Zbl 1263.11061; Zbl 1264.11044]. To find out a general theory of zeta functions and their variants occurring in various arenas such as Dirichlet series, Sato invented a theory of prehomogeneous vector spaces, to which Tatsuo Kimura contributed much, finally publishing a book on it [Zbl 1035.11060]. Masaki Kashiwara did computation in prehomogeneous vector spaces by using algebraic analysis [Zbl 0381.43005; Zbl 0456.58034]. The chapter is closed with a chronology of number theory beginning with the publication of Teiji Takagi's [JFM 34.0237.01] in 1903 and ending with Mochizuki's announcement of his alleged solution of the so-called \textit{$abc$ conjecture} in 2012.
Chapter 4, consisting of three sections, is concerned with mathematical physics. \S 1 addresses Huzihiro Araki and the \textit{theory of operator algebras}, characters on the scene being Huzihiro Araki, Alain Connes, Daniel Kastler, Rudolf Haag, Minoru Tomita, Masamichi Takesaki and Robert Powers.
Araki became a professor of RIMS in 1964 and retired in 1997.
\S 2 discusses Noboru Nakanishi and his quantum field theory. In 1955 Nakanishi as well as Huzihiro Araki entered the graduate school at Kyoto University to study physics under the direction of Hideki Yukawa, a Nobel prize laureate.
Nakanishi made his debut in the community of physicists by his \textit{topological formula of Feynman integrals} [Zbl 0077.21603], claiming that Feynman integrals are determined completely by the topology of Feynman diagrams. He has finally published a book [Noboru Nakanishi, Graph Theory and Feynman Integrals, Gordon \& Breach Science Pub., 1971]. Masaki Kashiwara and others addressed Feynman integrals from the standpoint of algebraic analysis in the 1970s [Zbl 0454.46034; Zbl 0392.46025; Zbl 0449.35095; Zbl 0527.58036; Zbl 0454.46035; Zbl 0385.35003].
During 1963--1965 Nakanishi stayed at Brookhaven National Laboratory, when he studied analyticity of scattering amplitudes and Bethe-Salpeter equations, which resulted in so-called \textit{Nakanishi-Lautrup formalism of abelian gauge theory} [Noboru Nakanishi, ``Ordinary and generalized Bethe-Salpeter equations in the unequal-mass case'', Phys. Rev. 147, 1153 (1966); Noboru Nakanishi, ``Covariant quantization of the electromagnetic field in the Landau gauge'', Progress of Theoretical Physics 35, 1111--1116 (1966); Noboru Nakanishi, ``Quantum electrodynamics in the general covariant gauge'', Progress of Theoretical Physics 38, 881--891 (1967)].
Nakanishi became a member of RIMS in 1966 and retired in 1997. In 1977, after Nakanishi-Lautrup formalism Taichiro Kugo and Izumi Ojima succeeded in canonically quantizing gauge fields [Zbl 1098.81591; Zbl 1098.81592]. Ojima studied mathematical physics under the direction of both Huzihiro Araki and Noboru Nakanishi after he graduated from the Faculty of Medicine at Kyoto University. It became Nakanishi's life work to quantize gravity after Nakanishi-Lautrup formalism [Zbl 1098.83558; Zbl 1098.83515; Zbl 1098.83517; Zbl 1098.83518; Zbl 1098.83561; Zbl 1074.83513; Zbl 1098.83559; Zbl 1098.83514; Zbl 1060.83515; Zbl 1060.83516; Zbl 1059.83511; Zbl 1074.83515; Zbl 1074.83514; Zbl 1059.83513; Zbl 1059.83512; Zbl 1098.83560; Zbl 1098.83562; Zbl 0545.53055; Zbl 0527.53048; Zbl 1046.83503; Zbl 0979.83508; Zbl 0979.83506; Zbl 0672.53069; Zbl 1058.83513; Zbl 1098.83516].
Both Nakanishi as well as Richard Feynman is strongly opposed against superstring theory, saying that superstring theory is of no coherent theory.
\S 3 is concerned with Hirosi Ooguri and \textit{superstring theory}. It was Mikio Sato, the then incumbent director of RIMS, that invited Ooguri to RIMS in 1990, saying that superstring theory is mathematically attractive and affluent, to say nothing of the problem whether it is really the theory of elementary particles. Ooguri left RIMS for the University of California at Berkeley in 1994. In the 1990s Ooguri and other three arrived at \textit{BCOV equations} [Zbl 0815.53082; Zbl 0908.58074; Zbl 0899.32008; Zbl 0919.58067], showing that topological string theory is useful in actual computation. In 2004 Ooguri and other two obtained the so-called \textit{OSV formula} [Hirosi Ooguri, Andrew Strominger and Cumrun Vafa, ``Black hole attractors and topological string'', Physical Review D (3) 70, 106007 (2004)], announcing a beautiful and highly unexpected proposal that the number of black hole states in certain string theories obtained by compactification on a Calabi-Yau manifold $X$ is to be expressed in terms of the topological string partition function of $X$, namely in terms of the so-called Gromov-Witten invariants of $X$, which has won the Eisenbud prize for 2008. In 2010 Ooguri and others [Zbl 1266.58008] claimed that the elliptic genius of the $K3$ surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$, suggesting that there is a sigma-model conformal field theory with $K3$\ target of $M_{24}$ symmetry. The chapter is concluded with chronologies of (1) operator algebras and Huzihiro Araki, (2) quantum field theory and Noboru Nakanishi, and (3) superstring theory and Hirosi Ooguri.
Chapter 5, being neither divided into sections nor accompanied by a chronology, is concerned with Kiyosi It么 and his theory of \textit{stochastic integration} and \textit{stochastic differential equations} known as \textit{It么 calculus}. Since he got married young on graduation from the University of Tokyo, he worked first for the Ministry of Finance and then for the Statistics Bureau of Japan, just as young Albert Einstein worked at the Federal Office for Intellectual Property in Bern, when he found out the theory of special relativity. The basic idea of It么 calculus which he encountered while working at the Statistics Bureau of Japan for five years was published in Gully printing [Kiyosi It么, Differential equations determining a Markov process, Zenkoku Sizyo Sugaku Danwakai-si (J. Pan-Japan Math. Coll.), 1942, 1352--1400]. It was written in Japanese and was translated into English to be published by American Mathematical Society [Zbl 0054.05803]. The development of It么 calculus can be seen in his [Zbl 0060.29105; Zbl 0063.02992; Zbl 0039.35103; Zbl 0045.07603; Zbl 0044.12202; Zbl 0049.08602; Zbl 0053.27302].
In 1943 It么 became an associate professor of the newly established Department of Mathematics at the Faculty of Science of Nagoya University, where he encountered other members
Sigekatu Kuroda [Zbl 0060.08903; Zbl 0061.05901],
Kiyoshi Noshiro [Zbl 0024.33002; Zbl 0021.23903],
Kosaku Yosida [Zbl 0023.39702; Zbl 0024.04201; Zbl 0024.21202] and
Tadasi Nakayama [Zbl 0061.04001; Zbl 0021.29402; Zbl 0019.10202].
In 1952, thanks to Yasuo Akizuki whose speciality is far away from that of It么 but who was enthusiastic over collecting excellent and challenging young mathematicians, It么 as well as Masayoshi Nagata moved from Nagoya University to Kyoto University. It么 stayed at the Institute for Advanced Study at Princeton for two years from 1954, where he got acquainted with a then graduate student Henry McKean of professor William Feller and finally published a book with him [Zbl 0285.60063]. In 1966 It么 left Japan for Aarhus University in Denmark, where he stayed until 1969 and then moved to Cornell University in USA. In 1975 It么 returned to RIMS in Japan, and was the incumbent director of RIMS from 1976 until he retired in 1979. In 1975 It么 invited Heisuke Hironaka from Harvard to RIMS. Here is a conversation between them on that occasion, showing great flexibility of personnel affairs at RIMS.
\begin{itemize}
\item It么: Now there is a vacancy on professorship at the Division of Nonlinearity at RIMS (the associate professor at that division was Takahiro Kawai and the assistant professor there was Michio Jimbo at that time).
\item Hironaka: I have studied algebra, algebraic geometry and commutative algebra, where only natural and simple objects are considered. I have never used even partial differentiation in my work.
\item It么: Don't worry. Even in algebraic geometry, polynomials are used. Polynomials are undoubtedly nonlinear functions in general, aren't they? So you are indeed versed in nonlinearity.
\end{itemize}
I became a graduate student at RIMS in April 1976, for that I had to pass a two-days examination in Summer 1975, the first day being a writing examination and the second day being an interview. Here is a conversation between It么 and me (Nishimura) on that interview.
\begin{itemize}
\item It么: Are there any sets which are not Lebesgue measurable?
\item Nishimura: Yes, of course. By cardinality argument, ...
\item It么: No, I want a constructive proof!
\item Nishimura: OK. Wait just for a moment. ...
\item It么: We have no time to wait. That is OK. As far as you are concerned, everything is good enough. Do you need a scholarship? ...
\end{itemize}
Chapter 6, consisting of three sections with a column, is concerned with applied mathematics.
\S 1 is concerned with computer science, centering on
Kyoto Common Lisp.
Characters on the scene are
Reiji Nakajima [Zbl 0354.02023; Zbl 0373.68025; Zbl 0432.68011; Zbl 0519.68003; Zbl 0595.68008],
Taiichi Yuasa [Zbl 0442.60093; Zbl 0625.68006; Zbl 0649.68003] and
Masami Hagiya [Zbl 0592.68032; Zbl 0522.03041; Zbl 0712.68057; Zbl 0873.68186].
It is regrettable that
Takeshi Hayashi [Zbl 0607.68060; Zbl 0602.68036; Zbl 0544.68042; Zbl 0499.68029; Zbl 0391.03012; Zbl 0319.68043],
Hirokazu Nishimura [Zbl 0401.03005; Zbl 0437.03034; Zbl 0423.68005; Zbl 1273.03089; Zbl 0574.51012],
Hiroakira Ono [Zbl 0281.02033; Zbl 0253.02022; Zbl 0246.02021; Zbl 0249.02022; Zbl 0226.02025],
Susumu Hayashi [Zbl 0618.54030; Zbl 0592.03010; Zbl 0513.68022; Zbl 0514.03035],
Masahiko Sato [Zbl 0444.03010; Zbl 0405.03013; Zbl 0405.03013; Zbl 0274.02029],
Takumi Kasai [Zbl 0301.68081; Zbl 0291.68005; Zbl 0289.68041; Zbl 0385.68047] should be missing here. All of them were once assistant professors at the division of Computer Science of RIMS.
\S 2 addresses fluid dynamics, in particular, \textit{Navier-Stokes equation}.
Characters on the scene are
Kanefusa Gotoh [Zbl 0112.19604; Zbl 0112.19603; Zbl 0095.21702; Zbl 0227.76056; Zbl 0226.76015],
Shigeo Kida [Zbl 0683.76036; Zbl 0673.76069; Zbl 0721.76041; Zbl 0712.76052; Zbl 0748.76064; Zbl 0758.76029; Zbl 1023.76557; Zbl 0939.76537; Zbl 0939.76554] and
Hisashi Okamoto [Zbl 0597.35104; Zbl 0596.76119; Zbl 0558.76111; Zbl 0678.76013; Zbl 0668.35006; Zbl 0617.35134; Zbl 0925.76104; Zbl 0850.76796; Zbl 0939.76513; Zbl 1306.76016; Zbl 0991.76526].
\S 3 deals with \textit{optimization}, centering upon \textit{discrete convex analysis}.
Characters on the scene are
Masao Iri [Zbl 0524.94033; Zbl 0451.90053],
Fujishige [Zbl 0571.90062; Zbl 0563.06010; Zbl 0665.90074; Zbl 0658.90062; Zbl 0770.90073; Zbl 0855.68107; Zbl 1296.90104; Zbl 1205.05237],
Kazuo Murota [Zbl 0639.05037; Zbl 0721.73021; Zbl 0711.68066; Zbl 0834.05037; Zbl 0938.65519; Zbl 1274.90528] and Satoru Iwata [Zbl 0838.05024; Zbl 0892.65027; Zbl 1296.90103; Zbl 1102.05048; Zbl 1193.05131; Zbl 1191.94167].
Chapter $\infty$ is concerned with the future of RIMS and that of mathematics itself. To do mathematics, we need nothing but enough time to concentrate. I became an assistant professor at RIMS in November 1979, when I enjoyed the following short conversation with Professor Satoru Takasu inviting me to assistant professorship at RIMS.
\begin{itemize}
\item Nishimura: What is the duty of an assistant professor at RIMS?
\item Takasu: You have only one duty, which is to get a salary every month. You should not forget your duty.
\end{itemize}
I was lucky to enjoy that position until March 1986. RIMS can afford every member, but professors affluent time for mathematics. The authors prove the hyperbolic Carnot theorem in the Poincar茅 disc model of hyperbolic geometry. They also prove the reverse triangle inequality, and the reverse M枚bius triangle inequality. | 0 |