Toshikazu Sunada

Toshikazu Sunada ( Japanese 砂 田 利 一 , Sunada Toshikazu ; born September 7, 1948 in Tokyo ) is a Japanese mathematician who, among other things, deals with geometric analysis and analysis on graphs.

Life

Sunada studied from 1968 at the Technical University of Tokyo (Tokyo Institute of Technology) among others Koji Shiga and at the University of Tokyo , where he received his diploma at Mikio Ise in 1974 (where he of Kunihiko Kodaira was tested) and 1977 doctorate. From 1974 he was also a researcher at the University of Nagoya (1975 to 1977 also at the University of Tokyo), where he became assistant professor in 1982 and professor in 1988. From 1991 he was a professor at the University of Tokyo and from 1993 at the University of Tōhoku , where he has retired since 2003. He has been a professor at Meiji University since 2003 . He is also at the Meiji Institute for Advanced Study in Mathematical Sciences in Tokyo. Among other things, he was visiting scholar at IHES (1988), the University of Bonn (1979/80) and the Max Planck Institute for Mathematics in Bonn (2008), the Humboldt University Berlin (2008), the Isaac Newton Institute in Cambridge (2007), the Institut Henri Poincaré in Paris, the Mittag-Leffler Institute , the Academy of Sciences in Beijing, the MSRI , the Tata Institute of Fundamental Research , in the Philippines and in Singapore.

He is the co-editor of a Japanese mathematics magazine called Have Fun With Mathematics (Nihon Hyoron-sha).

plant

He dealt with geometric analysis (especially spectral geometry), complex geometry (geometry of the functions of several complex variables), probability theory. In the mid-1980s he gave a general construction of isospectral manifolds, that is, those with the same spectrum of the Laplace operator. This was an important advance in the problem posed by Mark Kac of finding manifolds that are different despite the same spectrum ( Can one hear the shape of a drum? ). The problem was solved (using Sunada methods) in 1992 by Carolyn Gordon, Scott Wolpert, David Webb in a positive sense.

From Sunada comes a graph-theoretical interpretation of the Ihara zeta function (from Yasutaka Ihara ), with an explicit formula similar to that of Selberg's zeta function (only in this case with the eigenvalues ​​of the neighborhood matrix on the one hand and the lengths of closed cycles of the graph on the other) . He also proved that the Riemann hypothesis for the Ihara zeta function of a (connected k-regular) graph is equivalent to the statement that the graph is a Ramanujan graph. With Atsushi Katsuda he gave an analogue to Dirichlet 's theorem about prime numbers in arithmetic Consequences in the theory of dynamic systems, in the consideration of the density of closed orbits of Anosov rivers on compact manifolds to a certain homology class. Sunada studied the asymptotic behavior of random walks on crystal lattices. He also discovered a crystal lattice shape, the K4 crystal, which in its highly symmetrical behavior in terms of the equivalence of the orientations in space (isotropy) only with the diamond lattice in three dimensions and the honeycomb structure (hexagonal crystal lattice, realized in graphene ) in two dimensions is comparable.

In 1987 he received the Iyanaga Prize from the Japanese Mathematical Society . He was invited speaker at the ICM 1990 in Kyoto ( Trace formulas in spectral geometry , with M. Nishio).

literature

• Motoko Kotani, Hisashi Naito, Tatsuya Tate (editors): Spectral analysis in geometry and number theory, Contemporary Mathematics Vol. 484, American Mathematical Society (with biography of Sunada by Polly Wee Sy and Atsushi Katsuda), 2009, Conference on Sunada´s 60th birthday, 2007 at Nagoya University

Fonts

• with Peter Kuchment, Pavel Exner, Jonathan Keating, Alexander Teplyaev (editors): Analysis on Graphs and its applications, American Mathematical Society 2008, Proc. Symp. Pure Math. (Therein from Sunada: Discrete geometric analysis)
• with Koji Shiga: A mathematical gift III- the interplay between Topology, Functions, Geometry and Algebra, American Mathematical Society 2005

Individual evidence

1. ↑ Holomorphic equivalence problem of bounded Reinhardt domains , Mathematische Annalen 1978, as well as two other works Implicit function theorem for nonlinear elliptic operators , Random walks on a Riemann Manifold .
2. ↑ Sunada Riemannian coverings and isospectral manifolds , Annals of Math., Vol. 121, 1985, pp. 169-186.
3. ↑ Gordon, Wolpert, Webb One cannot hear the shape of a drum , Bulletin AMS, Vol. 27, 1992, p. 134.
4. ↑ every node has k neighbors.
5. ↑ A graph with a certain upper bound ( ) for the magnitudes of the eigenvalues ​​of its neighborhood matrix (if these are different from k), with the largest possible band gap in the spectrum, i.e. the distance between the magnitude of the eigenvalue, which is maximal but smaller than k, of the maximum possible eigenvalue amount k. Examples are cliques (in which every vertex is connected to every other) and Peterson graphs. They have applications in robust computer networks and in error-correcting codes ${\displaystyle 2{\sqrt {k-1}}}$
6. ↑ Katsuda, Sunada: Homology and closed geodesics in a compact Riemann surface, Amer. J. Math., Vol. 110, 1988, pp. 145-156, same: Closed orbits in homology classes, Publ. Math. IHES, Vol. 71, 1990, pp. 5-32 ( Memento of the original, July 4 2015 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. . 
7. ↑ Sunada, Kotani Spectral geometry of crystal lattices , Contemporary Mathematics, Vol. 338, 2003, p. 271, the same Albanese maps and off diagonal long time asymptotic of the heat kernel , Communications in mathematical physics, Vol. 209, 2000, p. 633.
8. ↑ already described by Fritz Laves in 1934
9. ^ Sunada Crystals that nature might miss creating , Notices AMS 2008 .