Mikio Satō

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Mikio Satō ( Japanese 佐藤 幹 夫 , Satō Mikio ; born April 18, 1928 in Tokyo ) is a Japanese mathematician who deals primarily with analysis, mathematical physics and is also known for a number theoretical conjecture.

Life

Satō was the son of a lawyer. During the Second World War, the family in Tokyo was bombed out. Satō himself was hauling coal in a factory during this time. From 1945 to 1948 he attended the 1st secondary school, which was then considered an elite school.

He then worked as a high school teacher to support his family and remained a teacher until 1958. He also studied from 1949 at the University of Tokyo . His written work received top marks, but since he had missed exams, he could not become an assistant and initially studied further, this time theoretical physics, among others with Shin'ichirō Tomonaga . In the summer of 1957 he wrote a thesis on the theory of hyperfunction in order to be accepted as a doctoral student in the mathematics faculty. Shōkichi Iyanaga made sure that he was hired as an assistant (actually he was assistant to Kōsaku Yoshida ) and in 1963 he received his doctorate in Tokyo. In 1960 he became a teacher at the Tokyo College of Education, and from 1960 to 1962 he was at the Institute for Advanced Study (Iyanaga had sent his work to André Weil ). He was then a professor at Osaka University and Tokyo University. In 1970 he became a professor at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University . From 1987 to 1991 he was director of the RIMS. He is currently Professor Emeritus at the University of Kyoto.

In 1969 he received the Asahi Prize and in 1976 the Prize of the Japanese Academy of Sciences . In 1984 he received the Japanese Order of Culture and in 1987 the Fujiwara Prize . Satō has been a member of the American National Academy of Sciences since 1993 . In 1997 he received the Rolf Schock Prize and in 2003 the Wolf Prize . In 1983 he gave a plenary lecture at the ICM in Warsaw ( Monodromy theory and holonomic quantum fields - a new link between mathematics and theoretical physics ) and in 1970 he was invited speaker at the ICM in Nice ( Regularity of hyperfunction solutions of partial differential equations ).

Masaki Kashiwara is one of his doctoral students . Another close collaborator in expanding micro-local analysis was Takahiro Kawai .

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Satō is best known for developing his theory of hyperfunctions , generalizations of distributions that are defined using the sheaf theory . If holomorphic functions are defined in the upper and lower complex half-planes, a hyperfunction can be defined as a difference on the real axis. It is invariant when adding a holomorphic function to and . He thus formulated a cohomological approach (without any boundary processes) to analysis parallel to Alexander Grothendieck at around the same time. From his work on hyperfunctions Satōs access to the micro-local analysis of partial differential equations (via the "analytical wavefront" of hyperfunctions) and to the algebraic theory of modules (developed by his student Kashiwara in his dissertation in 1969) developed. He already formulated the idea of ​​the analogy of modules via commutative rings to vector bundles via manifolds in a colloquium lecture in Tokyo in 1960. Satō was ahead of his time with his ideas: they seemed strange to analysts and were taken up relatively late or only in alternative formulations such as those of Lars Hörmander . The French mathematicians were an exception, where Satō's sheaf theoretical and algebraic approach came across prepared ground through the work of Leray , Cartan and Grothendieck.

Satō also worked on number theory. The until today unproven Satō Tate conjectures concern the fine distribution of the number of solutions of elliptic curves modulo and predict a statistical distribution function for the phases of the coefficients of the Hasse-Weil zeta functions of the curve that determine the fine distribution. In 1962 he also showed how the Ramanujan-Peterson conjectures about coefficients of modular forms follow from the Weil conjectures , later proven exactly by Pierre Deligne .

In 1970 he introduced Prehomogeneous Vector Spaces (PVS), finite dimensional vector spaces in which a subgroup of the general linear group of vector space has a dense open orbit. He applied it to number theory and classified irreducible PVS in 1977 with Tatsuo Kimura except for a transformation called castling .

Satō draws many of his motivations from physics. In mathematical physics he worked on soliton equations (partly with his wife Yasuko Satō), which he viewed as Graßmann manifolds of infinite dimensions. With his school he developed the direct method for solving soliton equations by Ryōgo Hirota in the 1980s , making connections to representations of infinitely dimensional Lie groups. With Tetsuji Miwa and Michio Jimbō he explicitly constructed the -point correlation functions in the two-dimensional Ising model with the help of the deformation theory (isomonodrome, that is, with preserved monodromic group) of common differential equations by Schlesinger from the 19th century.

Fonts

  • Theory of hyperfunctions. Volumes 1,2, Journal of the Faculty of Sciences, University of Tokyo, Vol. 8, 1959/60, pp. 139, 387.
  • with T. Kawai, M. Kashiwara: Microfunctions and pseudodifferential equations. In: Komatsu (Ed.): Hyperfunctions and pseudodifferential equations. Proceedings Katata 1971, Springer-Verlag, Lecture Notes in Mathematics Vol. 287, 1973, pp. 265-529.
  • The Hierarchy and infinite dimensional Grassmannian Manifolds. In: Theta Functions. Bowdoin 1985 Conference, Proceedings Symposia Pure Mathematics, Vol. 49, Part 1, AMS 1989, p. 51
  • with Yasuko Sato: Soliton equations as dynamical systems on infinite dimensional Grassmannian Manifolds. In: Nonlinear partial differential equations in applied sciences, Tokyo 1982. North Holland 1983, p. 259
  • with T. Miwa, M. Jimbo: Holonomic quantum fields. Part 1-5, Publications RIMS, Vol. 14, 1978, p. 223, Vol. 15, 1979, p. 201, 577, 871, Vol. 16, 1980, p. 531

Web links

Remarks

  1. ↑ independently from Lars Hörmander at about the same time , who then essentially dominated the development.
  2. ↑ independent in Russia Joseph Bernstein
  3. ↑ He got into analysis as an outsider. As a student of Takagi, his patron Iyanaga was actually a number theorist
personal data
SURNAME Sato, Mikio
ALTERNATIVE NAMES 佐藤 幹 夫 (Japanese)
BRIEF DESCRIPTION Japanese mathematician
DATE OF BIRTH April 18, 1928
PLACE OF BIRTH Tokyo