Title of Presentation

“Creation of mathematics: From number theory to geometry”

Nobody should have any objections against the view that polyhedra is the three-dimensional analogue of polygons, and that in this analogy the cube and the regular tetrahedron correspond to the square and the equilateral triangle, respectively. One may consider a similar correspondence in a different context; say, the three-dimensional analogue of the honeycomb lattice is the diamond lattice. In order to explain why they are thought of as being analogous, we need the idea of covering graphs.

As a matter of fact, there is another three-dimensional parallel to the honeycomb lattice in view of symmetry of crystal structures. The gist is that both honeycomb and diamond possess a special kind of symmetric property called strong isotropy. Another crystal structure with this property is what I designated the “diamond twin” (or K₄ crystal), which was discovered by the German crystallographer F. Laves in 1923 as a hypothetical crystal. In 2006, I found out that this structure deserves to be called the diamond twin.

History tells us that the threshold of class field theory due to D. Hilbert and T. Takagi was an analogy between the theory of covering surfaces and number theory. An analogy between the covering theory of Riemannian manifolds and number theory yielded an idea to construct a counter-example for the famous Kac’s problem “Can one hear the shape of a drum?” (C. Gordon, D. Webb, and S. Wolpert).

Furthermore we may consider a graph-theoretic analogue of Kronecker’s Jugendtraum (dream of youth) that originates in Abel’s work on solvable algebraic equations and is formulated as the problem of how to construct explicitly abelian extensions of a given number field. The outcome is applied to the design of crystal structure. It is also related to a discrete analogue of Abel-Jacobi maps associated with algebraic curves. Therefore, although the original dream of Kronecker has not yet been solved completely, one can somehow say that its geometric version was resolved.

As such, a resemblance to an already-known theory may give us a new idea or may lead us to a new theory. In this talk, while looking back my own studies based on analogy, I shall explain how mathematics is created.

Profile

A brief Biography

Born September 7, 1948 in Tokyo, currently Professor of mathematics at Meiji University, Tokyo, professor emeritus of Tohoku University, Tohoku, Japan, Deputy Director of Meiji Institute for Advanced Study of Mathematical Sciences, and Adjunct Professor at WPI Advanced Institute for Material Research, Tohoku University.

After my undergraduate studies at Tokyo Institute of Technology, I was admitted to the graduate school of the University of Tokyo (UT) and soon began my research under the supervision of Prof. Mikio Ise. Just after receiving master’s degree from UT in 1974, I was appointed as a research associate at Nagoya University (NU) where I was to stay for the next 15 years. Before joined Meiji University in 2000, I was professor of mathematics at Nagoya University (1988-1991), at the University of Tokyo (1991-1993), and at Tohoku University (1993-2003). In the meanwhile, I stayed for six months (1988) in Institut Hautes Etudes Scientifiques (IHES) as a guest professor, for a few months in Isaac Newton Institute at Cambridge as an organizer of a special project (2007), for seven months in Max Planck Institute in Bonn (2008) as a visiting professor, and for two months again in Newton Institute as a visiting fellow. I was involved in the creation of the School of Interdisciplinary Mathematical Sciences in Meiji University and is its first dean (2013-).

I was chosen a member of the Kyoto Prize Selection Committee for three terms (1989, 1994, 2002) in the past 20 years. In 2008, I was appointed a panel member of the European Research Council, an organization set up to promote outstanding, frontier research in all areas of science and humanities throughout Europe. My other services to the mathematics community include may two-term board membership of the Mathematical Society of Japan and the membership of the IMU-CDE committee where I served for two consecutive terms. Moreover, I helped in the organization of several major conferences, including the celebrated Taniguchi Symposia, held in Asia as a member of steering, scientific or advisory committee.

So far I have engaged in research on complex analysis, trace formulae, density of states, isospectral manifolds, twisted Laplacians, Ihara zeta functions, quantum ergodicity, discrete geometric analysis, strongly isotropic crystals (a diamond twin).

Details of selected Awards and Honors

the Iyanaga Prize of the Mathematical Society of Japan (MSJ) in 1987

Publication Prize of MSJ in 2013

A list of selected Publications

T. Sunada, Topological Crystallography – with a view towards Discrete Geometric Analysis -, Springer, 2012.

K. Shiga and T. Sunada, A Mathematical Gift, III, originally published in Japanese in 1996 by Iwanami Shoten, and translated by E. Tyler, Amer. Math. Soc., 2005.

T. Sunada, Discrete geometric analysis, Proceedings of Symposia in Pure Mathematics (ed. by P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev), 77 (2008), 51-86.

T. Sunada, Crystals that nature might miss creating, Notices of the AMS, 55 (2008), 208-215.

M. Kotani and T. Sunada, Albanese maps and off diagonal long time asymptotics for the heat kernel, Commun. Math Phys., 209 (2000), 633-670.

T. Sunada, Trace formulae in spectral geometry, Proc. ICM-90 Kyoto, Springer, 1991, 577-585.

T. Sunada and A. Katsuda, Closed orbits in homology classes, Publ. IHES. 71 (1990), 5-32.

T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985), 169-186.

J. Noguchi and T. Sunada, Finiteness of the family of rational and meromorphic mappings into algebraic varieties, Amer. J. Math. 104 (1982), 887-900.

T. Sunada, Holomorphic equivalence problem for bounded Reinhardt domains, Math. Ann. 235 (1978), 111-128.