Toshikazu Sunada

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Toshikazu Sunada
Born 1948 (age 65–66)
Tokyo, Japan
Nationality Japanese
Fields Mathematics (Spectral geometry and Discrete geometric analysis)
Institutions Nagoya University
Tokyo University
Tohoku University
Meiji University
Alma mater Tokyo Institute of Technology
Notable awards Iyanaga Award (1987) and Publication Prize (2013) of Mathematical Society of Japan

Toshikazu Sunada (砂田 利一 Sunada Toshikazu, born September 7, 1948) is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor of mathematics at Meiji University, Tokyo, and is also professor emeritus of Tohoku University, Tohoku, Japan. Before he joined Meiji University in 2003, he was professor of mathematics at Nagoya University (1988–1991), at the University of Tokyo (1991–1993), and at Tohoku University (1993–2003).

Main work

Sunada's work covers complex analytic geometry, spectral geometry, dynamical systems, probability, graph theory, and discrete geometric analysis. Among his numerous contributions, the most famous one is a general construction of isospectral manifolds (1985), which is based on his geometric model of number theory, and is considered to be a breakthrough in the problem proposed by Mark Kac in "Can one hear the shape of a drum?" (see Hearing the shape of a drum). Sunada's idea was taken up by C. Gordon, D. Webb, and S. Wolpert when they constructed a counterexample for Kac's problem. For this work, Sunada was awarded the Iyanaga Prize (1987) and Publication Prize (2013) of the Mathematical Society of Japan.

In a joint work with Atsushi Katsuda, Sunada also established a geometric analogue of Dirichlet's theorem on arithmetic progressions in the context of dynamical systems (1988). One can see, in this work as well as the one above, how the concepts and ideas in totally different fields (geometry, dynamical systems, and number theory) are put together to formulate problems and to produce new results.

His study of discrete geometric analysis includes a graph-theoretic interpretation of Ihara zeta functions, a discrete analogue of periodic magnetic Schroedinger operators as well as the large time asymptotic behaviors of random walk on crystal lattices. The study of random walk led him to the discovery of a "mathematical twin" of the diamond crystal out of an infinite universe of hypothetical crystals (2007). He named it the K4 crystal due to its mathematical relevance (see below).

For his work, see also Reinhardt domain, Ihara zeta function.

More on K4 crystal

It is believed that the crystallographer who explicitly described the network structure of the K4 crystal for the first time is Fritz Laves(1932). Since then, the structure has been rediscovered several times and also popularized by many people from various viewpoints. Consequently it has a number of names; say "Laves' graph of girth ten" (Harold Scott MacDonald Coxeter), "(10,3)-a" (A.Wells), "srs" (M. O'Keeffe), and "triamond" (John Horton Conway). The K4 crystal has a close relationship with the gyroid, an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970.

The K4 crystal as an abstract graph is the maximal abelian covering graph over the K4 graph, the complete graph with 4 vertices, while the diamond crystal is the maximal abelian covering graph over the graph with two vertices joined by 4 parallel edges (see covering space). Both crystals as networks in space are examples of “standard realizations”, the notion introduced in the study of random walks on general crystal lattices as a graph-theoretic version of Albanese maps (Abel-Jacobi maps) in algebraic geometry.

Moreover, the K4 crystal is a join of congruent decagonal rings. There are 15 decagonal rings passing through each vertex (atom). On the other hand, the diamond crystal has 12 hexagonal rings passing through each vertex. A big difference between K4 and diamond is that K4 has chirality while diamond does not have.

A remarkable fact pointed out by Sunada is that diamond and K4 have the "strong isotropy". Ordinarily, the isotropic property would describe that there being no distinction in any direction (note that the term "isotropic" is also used in a different context in crystallography). This strong isotropic property is the strongest one among all possible meanings of isotropy. In two-dimension, only honeycomb lattice (graphene) possesses this property. Actually, as proven by Sunada, the highly symmetric feature of the K4 crystal makes it the only mathematical relative of diamond and graphene.

Selected Publications by Sunada

  • T.Sunada, Holomorphic equivalence problem for bounded Reinhardt domains, Math. Ann. 235 (1978), 111-128
  • T.Sunada, Rigidity of certain harmonic mappings, Invent. Math. 51(1979), 297-307
  • 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, Riemannian coverings and isospectral manifolds, Ann. of Math. 121(1985), 169-186
  • T.Sunada, L-functions and some applications, Lect. Notes in Math. 1201(1986), Springer-Verlag, 266-284
  • A.Katsuda and T.Sunada, Homology and closed geodesics in a compact Riemann surface, Amer. J. Math. 110(1988), 145-156
  • T.Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology 28(1989), 125-132
  • A.Katsuda and T.Sunada, Closed orbits in homology classes, Publ. Math. IHES. 71(1990), 5-32
  • M.Nishio and T.Sunada, Trace formulae in spectral geometry, Proc. ICM-90 Kyoto, Springer-Verlag, Tokyo, (1991), 577-585
  • T.Sunada, Quantum ergodicity, Trend in Mathematics, Birkhauser Verlag, Basel, 1997, 175 - 196
  • M.Kotani and T.Sunada, Albanese maps and an off diagonal long time asymptotic for the heat kernel, Comm. Math. Phys. 209(2000), 633-670
  • M.Kotani and T.Sunada, Spectral geometry of crystal lattices, Contemporary Math. 338(2003), 271-305
  • T.Sunada, Crystals that nature might miss creating, Notices of the AMS, 55(2008), 208-215
  • 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
  • K.Shiga and T.Sunada, A Mathematical Gift, III, American Mathematical Society
  • T.Sunada, Lecture on topological crystallography, Japan. J. Math. 7(2012), 1-39
  • T. Sunada, Topological Crystallography, With a View Towards Discrete Geometric Analysis, Springer, 2013, ISBN 978-4-431-54176-9 (Print) 978-4-431-54177-6 (Online)

References

  • Atsushi Katsuda and Polly Wee Sy,, An overview of Sunada’s work
  • Meiji U. Homepage (Mathematics Department)
  • David Bradley, , Diamond's chiral chemical cousin
  • M. Itoh et al., New metallic carbon crystal, Phys. Rev. Lett. 102, 055703 (2009)
  • S.T. Hyde, M. O'Keeffe, and D.M. Proserpio, A short history of an elusive yet ubiquitous structure in chemistry, material, and mathematics, Angew. Chem. Int., 47(2008), 2-7
  • Diamond twin, Meiji U. Homepage
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