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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 geometric language of this vast work is largely that of \textit{A. Grothendieck} [Publ. Math., Inst. Hautes 脡tud. Sci. 4, 1--228 (1960; Zbl 0118.36206)] and while an acquaintance with his work is desirable for an understanding of the present paper almost all of his concepts which are used here are also defined. The proof of the resolution theorem follows the general lines of the proofs given by \textit{O. Zariski} [Ann. Math. (2) 45, 472--542 (1944; Zbl 0063.08361)] for the case of surfaces and threefolds, and so goes back to the early attempts of the italian geometers, especially \textit{B. Levi} [Torino Atti 33, 66--86 (1897; JFM 28.0557.01); Annali di Mat. (2) 26, 218--253 (1898; JFM 28.0557.02)]. However the present proof does not depend directly on results used in these earlier proofs. The local arguments require some knowledge of the theory of local rings, in particular the notions of the multiplicity of an ideal
in a local ring and of a regular system of parameters in a regular local ring.
Chapter 0 begins with a rapid summary of some of the basic concepts in the language of schemes, and a definition of the general notion of blowing-up in terms of a universal
mapping property. A proof of the existence theorem of blowing-up is outlined for certain categories. This leads to the definition and existence of monoidal transformations (and the centre of a monoidal transformation) in algebraic and analytic geometry. Let $B$ be a commutative ring with unity. A $\mathrm{Spec} (B)$-scheme $X$ is called an algebraic $B$-scheme if it is of finite type over $\mathrm{Spec} (B)$. A point $x$ of $X$ is said to be simple (muitiple) if the local ring $\mathcal O_{X,x}$, is (is not) regular. $X$ is said to be noningular if each point of $X$ is simple. The version of the resolution theorem (main theorem I) which is proved in the later chapters is formulated as follows. If $X (= X_0)$ is an algebraic $B$-scheme which is reduced and irreducible, where B is a field of characteristic zero, then there exists a finite sequence of monoidal transformations $f_i:X_{i+1}\to X_i$, $(i =0,1,\dots r-1)$ such that $X_r$ is non-singular and (a) the centre $D_i$ of $f_i$ is non-singular and (b) no point of $D_i$ is simple for $X_i$. Let $D$ be an algebraic subscheme of $X$ defined by the sheaf of ideals $\mathcal I$ on $X$, and let $\mathrm{gr}^p_D(X)$ be the quotient-sheaf $\mathcal I^p/\mathcal I^{p+1}$ restricted to $D$. $X$ is said to be normally flat along $D$ at a point $x$ of $D$ if the stalk of $\mathrm{gr}^p_D(X)$ at $x$ is a free $\mathcal O_{D,p}$-module for $p = 0,1,2,\dots X$ is said to be normally flat along $D$ if it is so at every point $x$ of $D$. It is natural to take the centre $D_i$ of the monoidal transformation $f_i$ to be equimultiple on $X_i$, the present proof of the resolution theorem imposes (c) $X_i$ is normally flat along $D$. It is shown later that (c) is in fact a stronger condition than equi-multiplicity. The other notion that plays an important role in the proof of the resolution theorem is that of normal crossings. Let $E$ be a reduced subscheme of a non-singular algebraic $B$-scheme $X$ which is everywhere of codimension one. $E$ is said to have only normal crossings at a point $x$ of $X$ if there exists a regular system of parameters $(z_1,\dots,z_n)$ of $\mathcal O_{X,x}$, such that the ideal in $O_{X,x}$ of each irreducible component of $E$ which contains $x$ is generated by one of the $z_i$, $E$ is said to have only normal crossings if it does so at every point of $X$. Running alongside the inductive proof of the resolution theorem is the inductive proof of the author's main theorem II which includes the following result. The complement of a Zariski open subset of a non-singular algebraic $B$-scheme (where $B$ is a field of characteristic zero) can be transformed by a finite sequence of monoidal transformations with non-singular centres into a subscheme which has only normal crossings.
Chapter 0 also contains a discussion of the analogues of main theorems I and II in the analytic case, and indicates how his results can be used to prove the resolution theorem for an arbitrary real analytic space. For a complex analytic space the passage from the local to the global resolution of singularities apparently introduces added difficulties, and in this case the author claims a proof of the corresponding theorem for a complex analytic space of dimension $\le 3$. The present methods make virtually no progress towards the resolution of singularities of algebraic $B$-schemes when $B$ is a field of positive characteristic.
Chapter I begins by restricting the ring $B$ to the class $\mathcal B$ of noetherian local rings $S$ with the properties (i) the residue field of $S$
has characteristic zero and (ii) if $A$ is an $S$-algebra of finite type and $\hat{S}$ denotes the completion of $S$ then, under the canonical morphism $\mathrm{Spec} (A\otimes_S \hat{S})\to\mathrm{Spec} (A)$, the singular locus of the former is the preimage of that of the latter spectrum. An algebraic $B$-scheme with $B$ in $\mathcal B$ is called an algebraic scheme. The two main theorems of Chapter 0 are reformulated in terms of two types of resolution data giving four fundamental theorems. The fundamental theorems are of two types; two of the theorems are separation theorems for the resolution data, while the other two are resolution theorems which imply the main theorems of Chapter 0. For technical reasons the fundamental theorems are concerned only with algebraic schemes which have a given irreducible non-singular ambient scheme; the resolution theorem without such an ambient
scheme is achieved by passing to the completion of a certain local ring, since every complete local ring in is a homomorphic image of a formal power series ring over a
field of characteristic zero.
Chapter II is a self-contained study of normal flatness, and is of an algebraic nature. The theorems proved when interpreted geometrically imply the following results, among others.
\begin{enumerate} \item[(1)] The set of points of a reduced subscheme $W$ of an algebraic scheme $V$ at which $V$ is normally flat along $W$ form an open dense subset of $W$.
\item[(2)] When $W$ is a non-singular irreducible subscheme of an algebraic scheme $V$, then for $V$ to be normally flat along $W$ it is necessary that $W$ should be equi-multiple on $V$.
\item[(3)] When $W,V$ are as in (2) and $V$ is embedded in a non-singular algebraic scheme $X$ such that the sheaf of ideals of $V$ on $X$ is locally everywhere generated by a single non-zero element, then for $V$ to be normally flat along $W$ it is necessary and sufficient that $W$ should be equi-muitiple on $V$.
\end{enumerate}
In Ch. III the local effect on singularities of permissible monoidal transformations is studied. Two local numerical characters are introduced as a measure of the severity of a singularity. In terms of these characters it is shown that a singularity is not made worse by any permissible monoidal transformation, and that any sequence of such transformations cannot make a singularity infinitely better. The aim of this local study is to prove the existence of a special coordinate system of the non-singular ambient scheme at a point and of a special base of the ideal defining the subscheme at the point, both of which have a certain stability with respect to the sequences of permissible monoidal transformations.
Chapter IV is devoted to the inductive proofs of the four fundamental theorems. Here the arguments are of a more geometric nature and rely on suitable geometric interpretations of the algebraic results in the two preceding chapters. In conclusion it is worth noting that the inductive proofs could not be carried out should the schemes be restricted to algebraic schemes over fields of characteristic zero; i.e. an essential part of the present proof of the resolution theorem for algebraic singularities is that the corresponding theorem for algebroid singularities should be proved at the same time. | 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 new approach is described to the evaluation of the S-matrix in three-dimensional atom-diatom reactive quantum scattering theory. A theory based on natural collision coordinates is developed, where the reaction coordinate can be viewed as fulfilling the same role as time in a time-dependent formulation. By writing the full wavefunction in coupled-channel form it is proved that the \(3D\) multi-channel quantum reactive scattering problem can be treated in the same way as an inelastic single-arrangement problem. In particularly in the work two type coupled-channel representations, which lead to to two different systems of coupled-channel differential equations. The first system of coupled-channel equations is solved with the help of the R-matrix propagation method yielding simultaneously the full wavefunction and all S-matrix elements without further calculation. The second one is treated similarly. In this way we avoid a great volume of grid computations for \(1D\) Schr枚dinger problem. The both algorithms use the intrinsic symmetry of scattering body-system, which allows to carry out maximally effective parallel computations of \(3D\) scattering problem. | 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 formalism of quantum gravity, it is known that the equal-time commutator between a tensor field and the \(B\) field \(b_ \rho\) is consistent with the rules of tensor analysis. Another tensorlike commutation relation is shown to exist for the equal-time commutator between a tensor and \(b_ \rho\), but at the same time its limitation is clarified. The quantum-gravity extension of the invariant \(D\) function is defined and proved to be affine-invariant. The four-dimensional commutation relation between a tensor and \(b_ \rho\) is investigated, and it is shown that the commutator consists of a completely tensorlike, manifestly affine-covariant part and a remainder, which is clearly distinguishable from the former. | 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. Similarly to atomic positions in a crystal being fixed, or at least constrained by the space group of that crystal, the displacements of atoms in a domain wall are determined or constrained by the symmetry of the wall given by the sectional layer group of the corresponding domain pair. The sectional layer group can be interpreted as comprised of operations that leave invariant a plane transecting two overlapping structures, the domain states of the two domains adhering to the domain wall. The procedure of determining the sectional layer groups for all orientations and positions of a transecting plane is called scanning of the space group. Scanning of non-magnetic space groups has been described and tabulated. It is shown here that the scanning of magnetic groups can be determined from that of non-magnetic groups. The information provided by scanning of magnetic space groups can be utilized in the symmetry analysis of domain walls in non-magnetic crystals since, for any dichromatic space group, which expresses the symmetry of overlapped structures of two non-magnetic domains, there exists an isomorphic magnetic space group. Consequently, a sectional layer group of a magnetic space group expresses the symmetry of a non-magnetic domain wall. Examples of this are given in the symmetry analysis of ferroelectric domain walls in non-magnetic perovskites. | 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. Prehomogeneous vector spaces were introduced by Mikio Sato in connection with a desire to understand when the Fourier transform of a complex power of a polynomial \(f\) is essentially again a complex power of some polynomial. The answer was found in terms of a big action of some group, with respect to which \(f\) is a relative invariant.
More specifically, a triplet \((G,\rho ,V)\) is called a prehomogeneous vector space over \(\mathbb C\) if \(\rho : G\to \text{GL}(V)\) is a rational representation of a connected algebraic group \(G\) on a finite-dimensional complex vector space \(V\), such that \(V\) has an orbit \(\rho (G)v_0\) dense in the Zariski topology (so that the orbit is a homogeneous space, and \(V\) is an ``almost'' homogeneous one). A relative invariant is a rational function \(f\) on \(V\), such that \(f(\rho (g)x)=\chi (g)f(x)\) for all \(x\in V,g\in G\), where \(\chi\) is a character of \(G\).
These notions are general enough to cover many applications, such as the derivation of functional equations and the study of special values for various zeta and \(L\)-functions in number theory. There are also connections with representation theory. The theory has been extended to the case of a non-Archimedean base field and was applied recently to \(p\)-adic Green functions [\textit{F. Sato}, Comment. Math. Univ. St. Pauli 51, 79--97 (2002; Zbl 1004.11067)] etc. On the other hand, there exists a deep classification theory of prehomogeneous vector spaces.
The book is intended as an introduction to the theory of prehomogeneous vector spaces containing all the algebraic and analytic preliminaries (Chapters 1 and 3), basic notions (Chapter 2), as well as a number of examples. However it reaches many deep results, such as the fundamental theorem on the Fourier transforms (in a rather general setting; Chapter 4), an extensive study of zeta functions on prehomogeneous vector spaces (Chapters 5 and 6) including the \(p\)-adic and adelic cases, and the classification of prehomogeneous vector spaces over \(\mathbb C\). An annotated bibliography provides instructions for further reading.
The book will surely have a wide readership among students and specialists in number theory, analysis, and representation theory. | 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 discuss a model for the two-dimensional hydroelastic time-evolution problem which describes the fluid-structure interaction. The elastic model accounts for membrane bending stresses and surface tension. The main result is devoted to the local well-posedness of the problem. The energy estimates in Sobolev spaces are used. | 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 0566.00032.]
We briefly review a fundamental theory of submodular and supermodular functions and consider a class of combinatorial problems described as those of minimizing submodular functions with possible constraints. Such problems are called submodular programs. We also consider the principal partition and the minimum-ratio problem from the point of view of submodular programs. | 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 classify extensions between finite irreducible conformal modules over a class of infinite Lie conformal algebras \(\mathfrak{B}(p)\) of Block type, where \(p\) is a nonzero complex number. We find that although certain finite irreducible conformal modules over \(\mathfrak{B}(p)\) are simply conformal modules over its Virasoro conformal subalgebra \(\mathfrak{Vir}\), there exist more nontrivial extensions between these conformal \(\mathfrak{B}(p)\)-modules. For extensions between other conformal modules, the situation becomes rather different. As an application, we also solve the extension problem for a series of finite Lie conformal algebras \(\mathfrak{b}(n)\) for \(n \geq 1\). | 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. The author discusses the finest block triangularization of nonsingular skew-symmetric matrices by simultaneous permutations of rows and columns. The hierarchical structure among components is represented in terms of signed posets. Using strongly connected component decomposition of bidirected graphs an algorithm to compute the finest block-triangular form efficiently is presented. | 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. Preliminary review/ Publisher's description \newline The book presents an original approach to the problem of the spatial-temporal chaos in deterministic systems. The basis is the notion of deal turbulence -- nonregular varying parameters of any medium without inherent resistence. The simplest mathematical patterns are given by difference equations with continuous argument and boundary value problems closed to difference equations. The special significance of the patterns is that these simulate many features of real turbulence such as cascade process of emergence of coherent structures, self-similarity and, finally, appearance of fractal objects, and, at the same time, can presently be completely investigated. One of the principal results is revealing self-stochasticity phenomenon when the attractor of deterministic systems contains random elements. The book combines analytical and qualitative methods with computer experiment and computer graphics. | 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 celebrated Mori theory says that any smooth projective complex threefold \(X\) can be transformed, by two kinds of birational operations called divisorial contractions and flips, to a model \(X'\) which has only ``very mild'' singularities and satisfies one of the following conditions:
(1) The canonical bundle \(K'\) of \(X'\) is nef; in this case \(\kappa(X)\geq 0\).
(2) There is a fibration \(f:X'\to Y\) such that \(\dim Y<3\) and \(-K'\) is \(f\)-ample; in this case \(X\) is uniruled.
The main aim of this paper is to generalize this theory to the case of families of threefolds. The deformation invariance of plurigenera of projective threefolds with mild singularities follows from this result. As another important application, it is shown that the ``birational moduli'' functor of threefolds of general type with a given Hilbert function is coarsely represented by a separated algebraic space of finite type. This theory applies also to the study of nonprojective deformation families of a projective threefold.
For the proof of the main result, it is shown that the two operations, divisorial contractions and flips, can be performed continuously for families. The contraction morphisms of extremal rays can be done continuously, so the problem is reduced to the existence of flips for families. The following result serves as a key lemma at this step: Let \(f:X\to Y\) be a proper bimeromorphic morphism of complex threefolds such that (1) \(X\) has only terminal singularities, (2) there is a normal point \(Q\) on \(Y\) such that \(C=f^{-1}(Q)\) is an irreducible curve and \(X- C\cong Y-Q\), (3) \(K_ XC<0\) for the canonical bundle \(K_ X\) of \(X\). Let \(t\) be a general element of the ideal \(I_ Q\) of \(Q\) in \(Y\) and let \(H'=\{t=0\}\) be its zero divisor. Then the singularity of \(H'\) at \(Q\) is classified very precisely. The result seems too technical to be reviewed here, but by using deformation theory of such surface singularities the authors deduce the existence of flips for families, where a deformation family of a threefold \(Y\) as above is regarded as the total space of a deformation family of the surface \(H'\).
The proof of this ``key lemma'' consists of very precise case-by-case analysis and numerous computations, and occupies most pages of this article. But it is really worth the pages spent. In fact, it establishes a classification theory of three dimensional small contractions and flips, with many by-products including the solution of Reid's conjecture about general elephants. Thus this paper deserves its title. | 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. Models of the consensus of the individual state in social systems have been the subject of recent research studies in the physics literature. We investigate how network structures coevolve with the individual state under the framework of social identity theory. Also, we propose an adaptive network model to achieve state consensus or local structural adjustment of individuals by evaluating the homogeneity among them. Specifically, the similarity threshold significantly affects the evolution of the network with different initial conditions, and thus there emerges obvious community structure and polarization. More importantly, there exists a critical point of phase transition, at which the network may evolve into a significant community structure and state-consistent 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. [For the entire collection see Zbl 0604.00007.]
We consider a free boundary problem for incompressibe perfect fluid with surface tension. The problem considered is as follows: A perfect fluid is circulating around a circle \(\Gamma\). The outward curve \(\gamma\) is a free boundary to be sought. We assume that the flow, which is confined between \(\Gamma\) and \(\gamma\), is irrotational. On the free boundary, surface tension works and makes the free boundary circular. On the other hand, the centrifugal force caused by the circulation of the flow makes the fluid go outward. Hence the balance of these two kinds of forces determines the geometrical properties of the free boundary. We show that there exist progressive waves, which are periodic motions of the fluid and are the exact solutions corresponding to the solitary waves. | 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 \(H\) be a compression body. A set of properly embedded arcs \(\tau\) in \(H\) is trivial if each arc is parallel to an arc in \(\partial_+ H\) or it is a vertical arc. Let \(T\) be a tangle in a compact orientable 3-manifold \(M\), that is, a collection of properly embedded arcs and circles in \(M\). Let \(\Sigma\) be a connected surface embedded in \(M\). We say that \(\Sigma\) is a bridge surface if \(\Sigma\) splits \(M\) into two compression bodies \(H^{+}\) and \(H^{-}\), it is transverse to \(T\), and \(\tau^{\pm}=H^{\pm}\cap T\) are trivial arcs in the corresponding compression body. For \(\Sigma\) a bridge surface, \(\chi(\Sigma)\) denotes the Euler characteristic of the punctured surface. In this paper the authors consider pairs of bridge splittings and want to determine bounds for a splitting that is obtained from both splittings via stabilizations and perturbations. In particular consider the splittings \((\Sigma, (H^{+},\tau^{+}),(H^{-},\tau^{-}))\) and \((\Sigma, (H^{-},\tau^{-}),(H^{+},\tau^{+}))\). Let \(\Sigma''\) be a surface that determines a splitting that is isotopic to stabilizations and perturbations of both splittings. In one of the main results of the paper, a lower bound for \(2-\chi(\Sigma'')\) is given, which depends on \(\chi(\Sigma)\) and \(d(\Sigma,T)\), the distance of \(T\) in the curve complex of \(\Sigma\). It is also shown that there exist infinitely many manifolds \(M_\alpha\) each containing a knot \(k_\alpha\), which has two bridge surfaces \(\Sigma\) and \(\Sigma'\) with \(\chi(\Sigma)=2s\) and \(\chi(\Sigma')=2s-2\), so that if \(\Sigma''\) is isotopic to stabilizations and perturbations of both surfaces, then \(\chi(\Sigma'')\leq 3s+4\). It follows also that for each \(n\geq 2\) there exists a knot \(K\) in \(S^3\) with bridge spheres \(\Sigma\) and \(\Sigma'\), with bridge numbers \(2n-1\) and \(2n\), so that a sphere that is isotopic to perturbations of both surfaces has at least \(3n-4\) bridges, i.e., for large \(n\) it requires many perturbations. These examples are analogous to the ones given by \textit{J. Hass}, \textit{A. Thompson} and \textit{W. Thurston} [Geom. Topol. 13, No. 4, 2029--2050 (2009; Zbl 1177.57018)], which are examples of Heegaard splittings that require many stabilizations to become equivalent. | 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 main objects of the paper under review are higher-dimensional analogues of two classical problems. The first one is the uniform boundedness conjecture (UB) stating that given a number field \(k\) and a positive integer \(g\), there exists a constant \(N=N(k,g)\) such that for any \(g\)-dimensional abelian \(k\)-variety the order of any its torsion \(k\)-point is at most \(N\) (this was proved by \textit{L.~Merel} [Invent. Math. 124, No. 1--3, 437--449 (1996; Zbl 0936.11037)] for \(g=1\)). A weaker variant, \(p\)-uniform boundedness conjecture (\(p\)UB), states that given \(k\), \(g\) as above and a fixed primed number \(p\), there exists a constant \(N=N(k,g,p)\) such that for any \(g\)-dimensional abelian \(k\)-variety \(A\) and any \(v\in A[p^{\infty}](k)\) the order of \(v\) is at most \(N\) (this was proved by \textit{Yu.~I.~Manin} [Izv. Akad. Nauk SSSR. Ser. Mat. 33, 459--465 (1969; Zbl 0191.19601)] for \(g=1\)). Both conjectures are widely open for \(g>1\).
The second one is the regular inverse Galois problem (RIGP): given a finite group \(G\) and a number field \(k\), does there exist a Galois extension \(L/k(T)\) with group \(G\) such that \(L\cap \bar k =k\)? An appropriate geometric reformulation of RIGP gives rise to the so-called modular tower conjecture (MT) of M.~Fried which, roughly, is a statement about the absence of rational points in certain Hurwitz spaces (moduli spaces of covers of curves).
The authors discuss higher-dimensional analogues of these two problems, (\(p\text{UB}_d\)) and (\(\text{MT}_d)\), respectively, where the field \(k\) is replaced with a \(d\)-dimensional \(k\)-scheme \(S\) of finite type. The main results of the paper state that both (\(p\text{UB}_1\)) and (\(\text{MT}_1\)) hold. Note that (\(p\text{UB}_1\)) in characteristic 0 as well as the implication \((p\text{UB}_1)\Longrightarrow (\text{MT}_1)\) in arbitrary characteristic were established in an earlier paper of the authors. | 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. Eine Zahlenfolge \(\{x_n\}\) hei脽t regul盲r, wenn es eine regul盲re Belegungsfunktion \(\varphi_x(u)\) (d. h. von beschr盲nkter Variation und normiert) gibt, so da脽
\[
x_n = \int\limits_0^1 u^n\, d\varphi_x(u) \qquad (n=0,1,2,\ldots)
\]
ist. Mit \(\{x_n\}\) ist die regul盲re Momentfunktion
\[
x(z) = \int\limits_0^1 u^z\,d\varphi_x(u)
\]
und die regul盲re momenterzeugende Funktion
\[
f_x(t) = x_0 - x_1t + x_2t^2 - \cdots = \int\limits_0^1 \dfrac{d\varphi_x(u)}{1+tu}
\]
assoziiert. Die aus \(\{x_n\}\) entspringende Hausdorff-Transformierte einer Folge \(\{s_n\}\) ist definiert durch
\[
t_m = \sum\limits_{n=0}^m C_{m,n}\varDelta^{m-n}x_n\cdot s_n \qquad (m=0,1,2,\ldots)
\]
mit
\[
C_{m,n} = \dfrac{m!}{n!(m-n)!} \;\text{ und } \;\varDelta^ix_j = x_j C_{i,1}x_{j+1} + C_{i,2}x_{j+2} - \cdots
\]
Die hierdurch sich ergebende Hausdorffsche Summationsmethode werde mit \([H, \varphi_x(u)]\) bezeichnet. Sind \(\{a_n\}\) und \(\{b_n\}\) zwei regul盲re Folgen und \(b_n\neq 0\), so ist nach Hausdorff dann und nur dann \([H, \varphi_a(u)]\supset [H, \varphi_b(u)]\) wenn die Folge \(\{a_n\}\) durch die Folge \(\{b_n\}\) ``teilbar'', d. h. \(a_n = b_nc_n\) ist, wo \(\{c_n\}\) eine regul盲re Folge darstellt. Unter dem Hausdorffschen Einschlie脽ungsproblem wird die Aufgabe verstanden, zu entscheiden, ob eine regul盲re Folge durch eine andere teilbar ist. Die L枚sungen, die in der vorliegenden Arbeit gegeben werden, sind eigentlich als eine Transformation des Problems anzusehen, indem sie die Frage nach der Existenz einer regul盲ren Folge \(\{c_n\}\) durch die nach der Existenz einer regul盲ren Belegungsfunktion \(\varphi_c(u)\) ersetzen. Es wird gezeigt, da脽 f眉r die Teilbarkeit von \(\{a_n\}\) durch \(\{b_n\}\) notwendig und hinreichend ist, da脽 eine regul盲re Belegungsfunktion \(\varphi_c(u)\) existiert derart, da脽
\[
f_a(t) = \int\limits_0^1 f_b(tu)\, d\varphi_c(u) \;\text{ f眉r } \;|t|<1
\]
ist. Ferner wird bewiesen, da脽 diese Bedingung 盲quivalent ist mit den fr眉her von \textit{Hille} und \textit{Tamarkin} (Proc, nat. Acad. Sci. USA 19 (1933), 573-577; JFM 59.0243.*) ohne Beweis angegebenen Bedingungen: Existenz einer regul盲ren Momentfunktion \(c(z)\), so da脽 \(a(z) = b(z)c(z)\) ist, bzw. einer regul盲ren Belegungsfunktion \(\varphi_c(u)\), so da脽 \(\varphi_a(v)=\int\limits_0^1\varphi_b\left(\dfrac{v}{u}\right)\, d\varphi_c(u)\) ist.
Setzt man speziell f眉r \([H, \varphi_b(u)]\) diejenige Hausdorffsche Methode, die mit der Ces脿roschen \((C, \alpha)\) 盲quivalent ist, d. h. diejenige mit
\[
\varphi_b(u) = \dfrac{1}{\varGamma (\alpha)}\int\limits_0^u \left(\log\dfrac{1}{t}\right)^{\alpha -1} dt, \quad \alpha > 0,
\]
so erh盲lt man Bedingungen f眉r \([H, \varphi_a(u)]\supset (C,\alpha)\).
Zum Schlu脽 wird folgende Frage behandelt: Wenn \(\{a_n\}\) und \(\{b_n\}\) 盲quivalente Hausdorffsche Methoden definieren, so soll entschieden werden, ob entsprechende Reihen in den Differenzmatrizen \((\Delta^ma_n)\) und \((\Delta^mb_n)\) auch 盲quivalente Hausdorffsche Methoden definieren. Dies wird speziell auf den Fall angewendet, da脽 die Ausgangsmethoden die H枚ldersche und die Cesar貌sche Methode \((H, \alpha)\) bzw. \((C, \alpha)\) sind. Werden die durch die \(m\)-ten Reihen in den Differenzmatrizen definierten Methoden mit \((H_m, \alpha)\) und \((C_m, \alpha)\) bezeichnet, so ergibt sich: \((H_m, \alpha)\supset (C_m, \alpha)\). Die umgekehrte Relation kann nur in einigen speziellen F盲llen mit der vorliegenden Methode gezeigt werden. | 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 stable marriage model due to Gale and Shapley is one of the most fundamental two-sided matching models. Recently, Fleiner generalized the model in terms of matroids, and Eguchi and Fujishige extended the matroidal model to the framework of discrete convex analysis. In this paper, we extend their model to a vector version in which indifference on preferences is allowed, and show the existence of a stable solution by a generalization of the Gale-Shapley algorithm. | 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 us recall that compact complex manifolds covered by noncompact Hermitian symmetric spaces share the same Chern numbers as the compact dual up to scaling. There have been established relations on all characteristic classes (not just the characteristic numbers) for normal projective connections following the Thomas theory of normal projective connections -- instead of Cartan's theory -- employing sheaves instead of principal bundles. This approach has been extended to obtain relations on characteristic classes for certain holomorphic \(G\)-structures. Cartan's theory generalizes easily to Cartan geometries, and manages abnormality without extra effort.
The author uses the Cartan theory to study relations between characteristic classes, generalizing the characteristic class results of all previously mentioned cases. The main result is:
Theorem. The ring of characteristic classes of a holomorphic Cartan geometry on a compact K盲hler manifold is a quotient of the ring of characteristic forms of the model via an explicit ring morphism.
The author also derives the relations on characteristic rings of various rational homogeneous varieties to give examples, and explains how to employ these results to study various types of complex analytic differential equations. | 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. Russian translation of [Lectures in modern analysis and applications. III, Lect. Notes Math. 170, 18-61 (1970; Zbl 0225.14022)]. | 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 paper studies when algebras with one operation form a monoidal category and analyzes Koszulness, cyclicity and (here introduced) dihedrality of the corresponding operads. The main tool, the polarization, permits to represent algebras with one operation without any symmetry (such as associative algebras) as structures with one commutative and one anticommutative operation (such as Poisson 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. The sixteen-dimensional Poincare-like superalgebra found previously in the manifestly covariant canonical formalism of quantum gravity is extended to a higher-dimensional one when matter fields involve a massless neutral scalar field and/or the electromagnetic field. The identities of Ward-Takahashi type and the consistency conditions are analyzed for the spontaneously broken symmetry generators of this extended superalgebra. We find that there is remarkable universality for the wave-function renormalization constants of the massless fields. | 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 extend Wiener's notion of `homogeneous' chaos expansion of Brownian functionals to functionals of a class of continuous martingales via a notion of iterated stochastic integral for such martingales. We impose a condition of `homogeneity' on the previsible sigma field of such martingales and show that under this condition the notions of purity, chaos representation property and the predictable representation property all coincide. | 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. Numerical simulations are used to study compressible turbulence with microscale Reynolds numbers up to 40 and rms Mach numbers M up to 0.9. The flows are randomly forced, with energy supplied to either the rotational or compressive components of kinetic energy, which is then transferred to internal energy through the pressure-dilatation interaction and viscous dissipation terms. Coupling between the two components of kinetic energy by the advection term is relatively weak, and most energy introduced to either component by the external force is transferred, without passing through the other component, to internal energy. A statistically quasiequilibrium of kinetic energy is realized while internal energy increases steadily. The spectral form of the rotational component of velocity, which hardly depends on M, is very close to that for incompressible flow. On the other hand, the compressive component depends strongly on M, especially at large wave numbers. | 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 investigate Carleson measures on the closure of the unit disc \(\mathbb{D}\), i.e., finite positive Borel measures \(\mu\) for which the formal identity \(J_\mu:H^p\to L^q(\mu)\) exists for given values of \(0<p<q<\infty\), as a bounded operator from the Hardy space \(H^p(\mathbb{D})\) into the Lebesgue space \(L^q(\mu)\). | 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. Nachdem Verf. in seiner gro脽en Arbeit ``脺ber eine Theorie des relativ-Abelschen Zahlk枚rpers'' (Journ. Coll. of Science T么kyo 41, Nr. 9; F. d. M. 47, 147, 1919-20) einen 脺berblick 眉ber s盲mtliche zu einem gegebenen K枚rper \(k\) relativ-Abelschen K枚rper gewonnen hatte, gelingt ihm in der vorliegenden Arbeit auf Grund jener Resultate verh盲ltnism盲脽ig leicht der Beweis des allgemeinen Reziprozit盲tsgesetzes. Denn der Hauptteil desselben ist in dem fr眉her bewiesenen Satze enthalten, da脽 die Zerf盲llung eines Primideals \(\mathfrak p\) aus \(k\) in einem relativ-Abelschen Oberk枚rper nur von der Klasse abh盲ngt, welcher \(\mathfrak p\) angeh枚rt bei einer Klasseneinteilung, die durch Kongruenzen nach der Relativdiskriminante definiert ist. Auch der Beweis von Takagi braucht aber wesentlich das Eisensteinsche Reziprozit盲tsgesetz, verl盲uft im 眉brigen aber naturgem盲脽 etwas anders als der Beweis von Furtw盲ngler, macht jedoch keinen Gebrauch von dem Hilbertschen Normenrestsymbol. | 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 full system which describes position and angular velocity of a rigid body is considered. By assuming that the attitude of the rigid body can be controlled by means of two independent torques, the authors address the problem of local asymptotic stabilizability at the origin.
It is well known that this system cannot be stabilized in the above sense by using time-invariant smooth feedback. In fact, this negative result remains true even if one allows discontinuous time-invariant feedback. On the other hand, early work seems to suggest that stabilizability can be performed by time-varying feedback. This can be actually proved to be true by virtue of certain non-constructive arguments and using the fact that the system is small time locally controllable.
Explicit constructions have been proposed in other papers. However, they work only in the particular case where the control torques act along the principal axes of inertia. The contribution of this paper consists in the construction of an explicit time-varying (periodic) almost-continuous feedback law which locally stabilizes the full system at the origin. Note that since the closed loop system is not continuous, solutions should be intended in Filippov's sense.
The first step of the construction is a change of coordinate which carries the system in the form
\[
\left\{\begin{aligned} \dot x_1&= x_5x_6+R_1(x),\\ \dot x^2&= x_1+cx_3x_6+R_2(x),\\ \dot x_3&= x_5+R_3(x),\\ \dot x_4&= x_6+R_4(x),\\ \dot x_5&=u_1,\\ \dot x_6&=u_2,\end{aligned} \right.
\]
where \(c\) is a positive constant. This system exhibits some interesting homogeneity properties with respect to a suitable dilation. In particular, as far as we are interested in local results, it is possible to prove that the terms \(R_1\), \(R_2\), \(R_3\), \(R_4\) are ``small'' and can be neglected.
The second step consists in the construction of a stabilizing feedback for the subsystem formed by the first three equations, where \(x_5\) and \(x_6\) are reviewed as inputs. The expression of the feedback law is explicitly given, and the proof is based on an (explicit) Lyapunov function.
Then the authors consider the system formed in the first, second, third, fifth and sixth equation as a cascade connection of the first three equations and two independent integrators. For this new system, a Lyapunov control function is explicitly constructed, starting with a homogeneous desingularizing procedure applied to the feedback law determined at the previous step. Hence, a new feedback is computed according to the well-known Artstein-Sontag universal formula.
Finally, the feedback law is modified in order to guarantee stability for the forth equation, as well. This can be done by exploiting the periodicity issue. For a certain time interval, the final feedback law coincides with the feedback defined at the previous step. For a subsequent time interval, we can take for instance many linear feedback which stabilizes the subsystem formed by the fourth and sixth equation.
Although it is not clear whether this construction may be feasible for practical applications, the paper is very interesting: it represents a very clever and elegant compound of different methodologies and approaches for stabilization of nonlinear systems. | 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. Modern modeling approaches in circuit simulation naturally lead to differential-algebraic equations (DAEs). The index of a DAE is a measure of the degree of numerical difficulty. In general, the higher the index, the more difficult it is to solve the DAE. The modified nodal analysis (MNA) is known to yield a DAE with index at most two in a wide class of nonlinear time-varying electric circuits.
In this paper, we consider a broader class of analysis method called the hybrid analysis. For linear time-invariant RLC circuits, we prove that the index of the DAE arising from the hybrid analysis is at most one, and give a structural characterization for the index of a DAE in the hybrid analysis. This yields an efficient algorithm for finding an optimal hybrid analysis in which the index of the DAE to be solved attains zero. Finally, for linear time-invariant electric circuits that may contain dependent sources, we prove that the optimal hybrid analysis by no means results in a higher index DAE than MNA. | 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. New results of the series of experimental and numerical investigations of the self-oscillatory regimes of plane vertical jet spouting from beneath the free surface of a heavy incompressible fluid in reservoirs of limited dimensions are presented. The experiments were performed on a rectangular-in-plan setup with near-bottom regime of fluid discharge. For several widths of the near-bottom orifices in the end walls of the setup the dependence of the self-oscillation period on the jet flow rate is studied. A considerable difference of these dependences from those earlier obtained in the case of the spouting with fluid discharge over a weir is found to exist. On certain ranges of control parameters it is established that the fountain self-oscillation periods are similar in value to those of natural oscillations of standing waves in the setup. A fairly narrow jet velocity range is revealed on which a hysteresis effect, that consists in the difference between the flow-rate-dependences of the self-oscillation period with gradual increase or decrease in the flow rate, is observable. The results of numerical calculations carried out using the STAR-CD software package are in good agreement with the experimental data. | 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 0684.00011.]
A route to chaos is investigated numerically in the motion of an incompressible viscous fluid which is governed by the Navier-Stokes equation with an external force. The high-symmetry is imposed on the velocity field. The complexity of the flow is characterized by the multi- periodicity of the temporal variation of the velocity field. By examining the frequency power spectrum of the energy and the orbit of the points of state we found the following scenario to chaos: steady \(\to\) simply periodic \(\to\) doubly periodic \(\to\) triply periodic \(\to\) non-periodic (chaotic) motions. The difference of the spatial complexity between the chaotic motion and the fully developed turbulence is discussed. | 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. At the end of the 16th century a revolution in navigation took place. According to the author, sailors in the past only coasted close to land, whereas after the revolution they used new techniques such as observing the sun and the stars, determining the latitude, and plotting charts.
This revolution was due to mathematicians whose aim it was to avoid the imperfections of navigation by means of new, mathematically founded methods. The most famous of these mathematicians was Henry Briggs; he never sailed himself but he was the one who invented many of the new methods. The author focuses on Briggs' close connections to the Virginia Company, the aims of which had been, first, to support colonization and, second, to find the Northwest Passage to the Pacific. Luke Foxe and Thomas James, famous captains of their time, used these new mathematical methods; they met each other in 1631 in Hudson Bay while they were searching for the Northwest Passage. Both of them were connected with Briggs: James had visited Briggs personally and Foxe had called some newly discovered Islands ``Brigges his Mathematicks''.
Unfortunately the reader does not get any impression of what this new kind of mathematics looked like. New tables, new instruments, and the usefulness of logarithms are discussed, but no detailed information is given. For this, see Thomas Sonar, Der fromme Tafelmacher. Die fr眉hen Arbeiten des Henry Briggs. Berlin: Logos Verlag (2002). | 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. An analysis of the data of a direct numerical simulation of a forced incompressible isotropic turbulence at a high Reynolds number (\(R_\lambda \approx 180\)) is made to investigate the interaction among three Fourier modes of wave numbers that form a triangle. The triad interaction is classified into six types according to the direction of the energy transfer to each Fourier mode:\( (+,+,-), (+,-,+), (+,-,-), (-,+,+), (-,+,-), (-,-,+)\), where the \(+\) (or \(-\)) denotes the energy gain (or loss) of the modes of the largest, the intermediate, and the smallest wave numbers in this order. The last three types of the interaction are very few. In the first type of the interaction, a comparable amount of energy is exchanged typically among three modes of comparable magnitude of wave numbers. In the second and third types, the magnitudes of the larger two wave numbers are comparable and much larger than the smallest one, and a great amount of energy is exchanged between the former two. This behavior of the triad interaction agrees very well with the prediction due to various quasinormal Markovianized closure theories of turbulence, and was observed before for lower Reynolds number turbulence by \textit{J. A. Domaradzki} and \textit{R. S. Rogallo} [Phys. Fluids A 2, 413--426 (1990)]. The fact that the dominant triad interactions involve different scales of motion suggests that the statistics of the small-scale motions of turbulence may be directly affected by the large-scale motions. Nevertheless, Kolmogorov's local energy cascade argument may hold at least partially because the energy is exchanged predominantly between modes of two comparable scales in the triad interaction. | 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. Describes the purpose of the Master of Education (M. Ed.) Program in Integrated Mathematics, Science, and Technology Education (MSAT Program) at The Ohio State University and discusses preservice teachers' attitudes and perceptions toward integrated curriculum. (Contains 35 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. Alternating finite automata on \(\omega\)-words are introduced as an extension of nondeterministic finite automata which process infinite sequences of symbols. The classes of \(\omega\)-languages defined by alternating finite automata are investigated and characterized under four types of acceptance conditions. It is shown that for one type of acceptance condition alternation increases the power in comparison with nondeterminism and for other three acceptance conditions nondeterministic finite automata on \(\omega\)-words have the same power as alternating ones. | 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 aim of this paper is to derive the existence of finite-dimensional attractors for reaction-diffusion systems with vanishing diffusion coefficients. Initial-boundary value problems are considered for the system
\[
(*)\quad \partial u/\partial t-d\Delta u+h(x,u)+f(x,u,v)=0,\quad \partial v/\partial t+\sigma (x)v+g(x,u)=0
\]
with \((u,v)\in \Omega \times {\mathbb{R}}^+\to {\mathbb{R}}^ 2,\) \(\Omega\) is an open bounded set of \(R^ n\), and for the system
\[
(**)\quad \partial u/\partial t-D\Delta u+f(x,u,v)=0,\quad \partial v/\partial t+G(x,u)v+g(x,u)=0
\]
with \(D=diag(d_ 1,...,d_{m_ 1})\), G is an \((m_ 2,m_ 2)\)-matix with positive definite symmetric part. The author proves the existence of an universal attractor in some function space and estimates the Hausdorff and fractal dimensions. He applies the results to some classical systems (Hodgkin-Huxley, Fitz Hugh-Nagumo, Field-Noyes). | 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 book, originally published in Japanese Iwanami Shoten, Tokyo 1986 and professionally translated into English, is a serious textbook whose clear ideas and specific details are based on the authors' experience in implementing Kyoto Common Lisp.
The first four chapters are a natural ``surface'' presentation of general and/or specific programming structures of (Common) Lisp. Chapters 5 to 9 come into a deeper representation of list structures and processing, functions, macros and declarations. Some delicate topics like the treatment of lexical closures, the notions and devices for generalized and special variables, function calls and mapping functions, dynamic binding, etc. are properly treated and illustrated. Chapters 10-12 are dealing with data types, input/output, efficient programming functions. Two useful appendices contain: (1) a text editor and a cross reference program, written in Common Lisp; (2) debugging techniques and tools.
A new and good book on Lisp, an (unnecessary) proving of the vitality of this ``old'' but ever younger language. | 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 new algorithm based on extension method and updated Newton method with global convergence for nonlinear iterative learning control problem is proposed. Since classical Newton-type iterative learning schemes are local convergent, conditions of local convergence can be hardly satisfied in practice. In order to widen the range of convergence, an extension method is introduced to iterative learning control problems. A new Newton-type iterative learning control scheme based on homotopy extension is presented, in which the initial control can be chosen arbitrarily. The solving process is subdivided to \(N\) subproblems by the new algorithm. The exchange column update Newton method is employed to solve the subproblems by a simple recurrent formula. Sufficient conditions for global convergence of this algorithm are given and proved. The implementation of the new algorithm has advantages of guaranteeing global convergence and avoiding complex calculation for nonlinear iterative learning control. | 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 logical system of inference rules intended to give the foundation of logic programs is presented. The distinguished point of the approach taken here is the application of the theory of inductive definitions, which allows us to uniformly treat various kinds of induction schemata and also allows us to regard negation as failure as a kind of induction schema. This approach corresponds to the so-called least fixpoint semantics. Moreover, in our formalism, logic programs are extended so that a condition of a clause may be any first-order formula. This makes it possible to write a quantified specification as a logic program. It also makes the class of induction schemata much larger to include the usual course-of-values inductions. | 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 automatic sign language translation, one of the main problems is the usage of spatial information in sign language and its proper representation and translation, e.g. the handling of spatial reference points in the signing space. Such locations are encoded at static points in signing space as spatial references for motion events.
We present a new approach starting from a large vocabulary speech recognition system which is able to recognize sentences of continuous sign language speaker independently. The manual features obtained from the tracking are passed to the statistical machine translation system to improve its accuracy. On a publicly available benchmark database, we achieve a competitive recognition performance and can similarly improve the translation performance by integrating the tracking features. | 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 sixteen-dimensional superspace structure of the manifestly-covariant canonical formalism of quantum gravity is investigated in detail. The field equations and the equal-time (anti-) commutation relations are shown to be expressible in the sixteen-dimensional form. The sixteen-dimensional Poincar茅-like superalgebra found previously is rederived in a unified way. It is shown that this superalgebra is simulated by a system of an idealized angular supermomentum and an idealized supermomentum, where the former is expressed as an antisymmetric product of the latter and an idealized position supervector. | 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 module \(M\) is extensionless if every extension of \(M\) by \(M\) splits. Let \(D\) be a countable Dedekind domain with quotient field, \(K\). Let \(M\) be a \(D\)-module with \(k\otimes_ D M\) an \(n\)-dimensional \(K\)-vector space, i.e. \(M\) is of finite rank, \(n\). We show that \(M\) is extensionless if and only if \(M\) is isomorphic to \(\bigoplus_ nR\), where \(R\) is a rank one \(D\)-module with \(R/D\) a direct sum of Pr眉fer modules (possibly none) and a finitely-generated torsion \(D\)-module. In order to prove this result we first prove that if \(M_ 1\) and \(M_ 2\) are torsion-free \(D\)- modules of finite rank, then \(\text{Ext}(M_ 2,M_ 1)\) is either 0 or has cardinality \(2^{\aleph_ 0}\). Analogous results are also proved when \(D\) is replaced by \(S\), \(S\) a countable finite-dimensional hereditary tame algebra over an algebraically closed field. | 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 formalism of quantum gravity, an expression is proposed for the four-dimensional (anti-)commutator between two sixteen-dimensional supercoordinates. The geometric commutation relation proposed previously is further studied; nongeometric parts, which arise owing to the quantum nature of the metric, are explicitly calculated for various fundamental fields and for \(E_ {\mu\nu}\), the nonclassical part of the quantum Einstein equation. | 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 study the interaction of small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbour interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intermediate values of time these solutions describe the interaction of two counter-propagating pulses. These solutions are stable with respect to perturbations in \(\ell^{2}\) and asymptotically stable with respect to perturbations which decay exponentially at spatial \(\pm \infty \). | 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 XIX the gravitational extension, \({\mathcal D}(x,y)\), of the Pauli-Jordan invariant \(D\) function is investigated in detail in both classical and quantum frameworks. Almost all its important properties are shown to be derivable even if an extra assumption \({\mathcal D}(y,x)=-{\mathcal D}(x,y)\) is removed. | 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. Several authors have proposed generalizations of the standard test for lack of fit in regression when replicate data is not available. In the present paper a more powerful test is developed using a model expansion based on splined ruled surfaces. Monte Carlo experiments show the gain of power of the ``spline'' test, and applications are presented to demonstrate how fitting the splined surfaces to data can be more easily achieved by reparameterizing the alternative model. | 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 paper under review deals with \(p\)-adic Galois representations. It uses the tool of almost 猫tale extensions. For Galois groups of local fields this goes back to \textit{J. T. Tate} [in: Proc. Conf. local Fields, NUFFIC Summer School Driebergen 1966, 158--183 (1967; Zbl 0157.27601)], and \textit{S. Sen} [Invent. Math. 94, No. 1, 1--12 (1988; Zbl 0695.12009)] has derived a description of semilinear Galois representations via modules with an endomorphism. The reviewer has extended this theory to schemes of higher dimension. The local theory follows from the almost purity theorem, and globalisation poses interesting new problems. The name Simpson correspondence stems from the well-known complex analytic analogue.
The authors give a very detailed introduction to the theory, smoothing out some difficulties by introducing new concepts. They also correct some mistakes of the reviewer. | 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. Review of [Zbl 1334.74001]. | 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 bisubmodular polyhedron is defined in terms of a so-called bisubmodular function on a family of ordered pairs of disjoint subsets of a finite set. We examine the structures of bisubmodular polyhedra in terms of signed poset and exchangeability graph. We give a characterization of extreme points together with an \(O(n^2)\) algorithm for discerning whether a given point is an extreme point, where \(n\) is the cardinality of the underlying set, and we assume a function evaluation oracle for the bisubmodular function. The algorithm also determines the signed poset structure associated with the given point if it is an extreme point. We reveal the adjacency relation of extreme points by means of the Hasse diagrams of the associated signed posets. Moreover, we investigate the connectivity and the decomposition of a bisubmodular system into its connected components. | 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 Markov process is constructed in an explicit way for an infinite system of entities arriving in and departing from a habitat \(X\), which is a locally compact Polish space with a positive Radon measure \(\chi\). Along with its location \(x \in X\), each particle is characterized by age \(\alpha \geq 0\)-time since arriving. As the state space one takes the set of marked configurations \(\hat{\Gamma}\), equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator \(L\), expressed through the measure \(\chi\) and a departure rate \(m (x, \alpha) \geq 0\), and acting on bounded continuous functions \(F:\hat{\Gamma} \rightarrow R\). For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero. | 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 author works out the concept of a split decomposition of a distributive lattice fitting into the general decomposition theory of Cunningham and Edmonds. He clarifies the amount of uniqueness valid for repeated decompositions, the structure of the building stones, and the reconstruction process. | 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 study \(\mathcal{N} = {2}\) supersymmetric \(\operatorname{SU}(2)\) gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional \(\operatorname{A}_{1}(2,0)\) theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the {\(\Omega\)}-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities. | 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 model in quantum field theory usually has several parameters, and it is important to understand how various quantities one can compute in such a model depending on the parameters. In the case of the nonlinear sigma-model, the moduli of the target space can be regarded as such parameters. Thus, we are interested in how amplitudes in the nonlinear sigma-model behave as we change the geometry of the target space.
In the case of the \(N= 2\) supersymmetric sigma-model in two dimensions, and when the target space is a Ricci-flat manifold (Calabi-Yau manifold) \(M\), the model becomes ultra-violet finite and the \(N= 2\) superconformal algebra is realized as symmetry of the model. One can make topological string theory from such a model''. | 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 studies the algebras \(D^n(K)\) of differentiable functions (up to the order \(n\)) on perfect planar compacta, thus continuing research done by \textit{W. J. Bland} and \textit{J. F. Feinstein} [Stud. Math. 170, No. 1, 89--111 (2005; Zbl 1077.46017)] and \textit{H. G. Dales} and \textit{J. F. Feinstein} [Indian J. Pure Appl. Math. 41, No. 1, 153--187 (2010; Zbl 1226.46053)].
Investigated are completeness and completion properties. Results on semi-simplicity are included, too. Unlike in the above mentioned publications, the author's approach is an operator theoretic one. Hoffmann's paper is well written and recommended to anyone interested in function 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. The distortion by large-scale random motions of small-scale turbulence is investigated by examining a model problem. The changes in energy spectra, velocity and vorticity moments, and anisotropy of small-scale turbulence are calculated over timescales short compared with the timescale of small-scale turbulence by applying rapid distortion theory with a random distortion matrix for different initial conditions: irrotational or rotational, and isotropic or anisotropic large-scale turbulence with or without mean strain, and isotropic or anisotropic small-scale turbulence.
We have obtained the following results: (1) Irrotational random strains broaden the small-scale energy spectrum and transfer energy to higher wavenumbers. (2) The rotational part of the large-scale strain is important for reducing anisotropy of turbulence rather than transferring energy to higher wavenumbers. (3) Anisotropy in small-scale turbulence is reduced by large-scale isotropic turbulence. The reduction of anisotropy of the velocity field depends on the initial value of the velocity anisotropy tensor of the small-scale velocity field \(u_ i\) defined by \(\overline{u_ iu_ j}/\overline{u_ lu_ l}-1/3\delta_{ij}\), and also on the anisotropy of the distribution of the energy spectrum in wavenumber space. The reduction in anisotropy of the vorticity field \(\omega_ i\) depends only on the vorticity anisotropy tensor. (4) The pressure-strain correlation is calculated for the change in Reynolds stress of the anisotropic small-scale turbulence. The correlation is proportional to time and depends on the difference between the velocity and wavenumber anisotropy tensors. These results (which are exact for small time) differ significantly from current turbulence models.
(5) The effect of large-scale anisotropic turbulence on isotropic small- scale turbulence is calculated in general. Results are given for the case of axisymmetric large scales and are compared with the observed behavior of small-scale turbulence near interfaces. (6) When a mean irrotational straining motion is applied to turbulence with distinct large-scale and small-scale components in their velocity field, the large-scale irrotational motions combine with the mean straining to increase further the anisotropy of the vorticity of the small scales, but the large-scale rotational motions reduce the small-scale anisotropy. For isotropic straining motion, the latter is weaker than the former. After the mean distortion ceases, both kinds of large-scale straining tend to reduce the anisotropy. This also has implications for modelling the rate of reduction of anisotropy. | 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 \(\mathbb{F}_q\) be the finite field of characteristic \(p\) with \(q\) elements and \(\mathbb{F}_{q^n}\) its extension of degree \(n\). We prove that there exists a primitive element of \(\mathbb{F}_{q^n}\) that produces a completely normal basis of \(\mathbb{F}_{q^n}\) over \(\mathbb{F}_q\), provided that \(n = p^\ell m\) with \((m, p) = 1\) and \(q > m\). | 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 \(U\) be a smooth and geometrically connected hyperbolic curve over a finite field \(k\). Grothen\-dieck's philosophy of anabelian geometry states that the isomorphism type of \(U\) as a scheme should be encoded in the isomorphism type of \(\pi^{\text{茅t}}_1(U)\) as a profinite group. This was proved to be true by Tamagawa (in the affine case) and Mochizuki (in the proper case). However, at present there is no example of a curve \(U\) for which the structure of \(\pi_1(U)\) is known. Hence, it is natural to search for better understood quotients of \(\pi_1(U)\) that still encode the isomorphy type of \(U\). The first half of the present paper is a survey of recent results concerning this search, while the final section presents a new proof of the authors' birational prime-to-characteristic result, which they established by different means in their previous paper [[1]: Publ. Res. Inst. Math. Sci. 45, No. 1, 135--186 (2009; Zbl 1188.14016)].
Section 1 of the paper reviews the Isom-form of Grothendieck's anabelian conjecture, which states that any isomorphism between 茅tale fundamental groups of hyperbolic curves over finite fields arises from a unique isomorphism of schemes. As mentioned above, this conjecture was proved by \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)] and \textit{S. Mochizuki} [J. Math. Kyoto Univ. 47, No. 3, 451--539 (2007; Zbl 1143.14305)] in the affine and proper cases, respectively. This result implies the birational analogue concerning isomorphisms between Galois groups of function fields, due originally to \textit{K. Uchida} [Ann. Math. (2) 106, 589--598 (1977; Zbl 0372.12017)].
After stating these older results, the authors describe their improvements in [1], where the full fundamental groups (resp. Galois groups) are replaced by the geometrically prime-to-characteristic quotients. Finally, they present a further refinement of these theorems to the case of geometrically pro-\(\Sigma\) fundamental groups (resp. Galois groups), where \(\Sigma\) is a sufficiently large set of primes distinct from the characteristic. Here ``sufficiently large'' is defined by a technical condition involving the non-injectivity of certain Galois-representations, but as a simple example, all cofinite sets of primes are ``sufficiently large''. At the other extreme, the authors stress the importance of investigating the case of \(\Sigma=\{l\}\), i.e. the question of whether the Isom-form of the Grothendieck conjecture holds for the geometric pro-\(l\) quotients, where \(l\) is a prime distinct from the characteristic.
Section 2 is devoted to a discussion of the difficulties in proving a Hom-form of the Grothendieck conjecture, which states that any continuous open homomorphism between 茅tale fundamental groups of hyperbolic curves over finite fields arises from a unique generically 茅tale morphism of schemes. These difficulties center on the lack of a good ``local theory'' for such homomorphisms. Motivated by these reflections, the authors describe a restricted class of homomorphisms with good local properties and then a recently obtained theorem in the birational case (resp. conjecture in the proper case) asserting the existence of a Hom-form for this restricted class of homomorphisms between Galois groups (resp. between fundamental groups).
The final section of the paper (Section 3) gives a new proof of the prime-to-characteristic version of Uchida's birational theorem mentioned in Section 1. In [1], this result was derived as a corollary to the prime-to-characteristic version of the Tamagawa-Mochizuki Isom-form for anabelian curves; here the result is proved directly using class field theory in a manner inspired by Uchida's original proof. | 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 this note we describe Rubk's latest puzzle, Rubik's Clock. Since the solution provides a nice application of linear algebra to a practical problem, we also provide a solution (which incidentally, is not provided at the time of purchase) to the puzzle.
There are \(12^{18}\) possible configurations of the 18 clocks, although as noted in the paper, not all of these are realizable. For each of the 16 settings of the four buttons, the four wheels can be moved in \(12^4\) ways. Hence there are \((12^4)^{16}= 12^{64}\) possible moves or actions that could be applied. Although these commute with each other, one might still expect the solution to Rubik's Clock to be very complex. However, as is shown, the solution is mathematically very simple, in fact much simpler than the group theoretical solution of Rubik's Cube. | 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 discuss how to derive the identities of Ward-Takahashi type corresponding to spontaneously broken symmetries in the manifestly covariant canonical formalism of quantum gravity. Based on the sixteen-dimensional Poincare-like superalgebra, it is found that various consistency conditions are derived for the space-time integrals of certain special vacuum expectation values of operator \(T\)-products. We also find that a relation holds for the wave-function renormalization constants | 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 exact asymptotic analysis is presented for the stress fields near the tip of a static plane stress mode II crack in perfectly plastic material. The solution obtained completes the class of the solutions known for static plane strain and plane stress mode I cracks and a plane strain mode II crack in the perfectly plastic material. However, the stress field near a mode II crack tip under plane stress in the perfectly plastic material seems to have not been obtained yet. It is shown that there are three discriminated sectors in the vicinity of the crack tip where the stress field is determined by the different formulae. They are the two sectors which can be described in slip line terminology as ``constant stress'' and ``centered fan'' sector of non-constant stress. These three different sector types constitute the complete set of asymptotically admissible solutions near the crack tip. The angles separating the sectors are numerically found from the solution of the system of the two trigonometric equations expressing the stress continuity conditions. The solution can not be considered as the full solution for the elastic -- perfectly plastic material since only the equilibrium equations and the von Mises yield criterion are analyzed. The solution will be full if it will be possible to analyze the deformation filed near the crack tip. In the framework of the present study it is possible to establish the strain asymptotic of the type \(1/r\) in the ``centered fan'' only. | 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 indefinite-metric quantum field theory of general relativity proposed previously, the explicit expressions for the Poincar茅 generators are derived on the basis of the canonical energy-momentum tensor and the canonical angular-momentum tensor. It is proved that the Poincar茅 generators have the right commutation relations with field operators, form the Poincar茅 algebra and are BRS-invariant.
Some speculation is made on what happens if space-time is not asymptotically flat. | 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 the context of the teleparallel equivalent of general relativity, we obtain the tetrad and the torsion fields of the stationary axisymmetric Kerr spacetime. It is shown that, in the slow rotation and weak-field approximations, the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field, and is thus responsible for the Lense-Thirring effect. | 0 |