is a Japanese

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

and author of many books and essays on mathematics and mathematical sciences. He is professor emeritus of both

Meiji University , abbreviated as Meiji (明治) or Meidai (明大'')'', is a private research university located in Chiyoda City, the heart of Tokyo, Japan. Established in 1881 as Meiji Law School (明治法律学校, ''Meiji Hōritsu Gakkō'') by three Meiji-er ...

and

Tohoku University , or is a Japanese national university located in Sendai, Miyagi in the Tōhoku Region, Japan. It is informally referred to as . Established in 1907, it was the third Imperial University in Japan and among the first three Designated National ...

. He is also distinguished professor of emeritus at Meiji in recognition of achievement over the course of an academic career. Before he joined Meiji University in 2003, he was professor of mathematics at

Nagoya University , abbreviated to or NU, is a Japanese national research university located in Chikusa-ku, Nagoya. It was the seventh Imperial University in Japan, one of the first five Designated National University and selected as a Top Type university of T ...

(1988–1991), at the

University of Tokyo , abbreviated as or UTokyo, is a public research university located in Bunkyō, Tokyo, Japan. Established in 1877, the university was the first Imperial University and is currently a Top Type university of the Top Global University Project by ...

(1991–1993), and at Tohoku University (1993–2003). Sunada was involved in the creation of the School of Interdisciplinary Mathematical Sciences at Meiji University and is its first dean (2013–2017). Since 2019, he is President of Mathematics Education Society of Japan.

Main work

Sunada's work covers

complex analytic geometry In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) Reduced ring, reduced or complex analytic space i ...

spectral geometry Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian m ...

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, discrete geometric analysis, and mathematical crystallography. 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 Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, and is considered to be a breakthrough in the problem proposed by

Mark Kac Mark Kac ( ; Polish: ''Marek Kac''; August 3, 1914 – October 26, 1984) was a Polish American mathematician. His main interest was probability theory. His question, " Can one hear the shape of a drum?" set off research into spectral theory, the i ...

in "Can one hear the shape of a drum?" (see

Hearing the shape of a drum To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" is the title of a 1966 article ...

). Sunada's idea was taken up by

Carolyn S. Gordon Carolyn S. Gordon (born 1950) is a mathematician and Benjamin Cheney Professor of Mathematics at Dartmouth College. She is most well known for giving a negative answer to the question "Can you hear the shape of a drum?" in her work with David Webb ...

, David Webb, and Scott A. Wolpert when they constructed a counterexample for Kac's problem. For this work, Sunada was awarded the Iyanaga Prize of the

Mathematical Society of Japan The Mathematical Society of Japan (MSJ, ja, 日本数学会) is a learned society for mathematics in Japan. In 1877, the organization was established as the ''Tokyo Sugaku Kaisha'' and was the first academic society in Japan. It was re-organized ...

(MSJ) in 1987. He was also awarded Publication Prize of MSJ in 2013, the Hiroshi Fujiwara Prize for Mathematical Sciences in 2017, the Prize for Science and Technology (the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology) in 2018, and the 1st Kodaira Kunihiko Prize in 2019. In a joint work with Atsushi Katsuda, Sunada also established a geometric analogue of

Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is als ...

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 function In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. The Ihara zeta function was first ...

s, a discrete analogue of periodic magnetic Schrödinger operators as well as the large time asymptotic behaviors of

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

on crystal lattices. The study of random walk led him to the discovery of a "mathematical twin" of the

diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of car ...

crystal out of an infinite universe of hypothetical crystals (2005). He named it the K₄ crystal due to its mathematical relevance (see the linked article). What was noticed by him is that the K₄ crystal has the "strong isotropy property", meaning that for any two vertices ''x'' and ''y'' of the crystal net, and for any ordering of the edges adjacent to ''x'' and any ordering of the edges adjacent to ''y'', there is a net-preserving congruence taking ''x'' to ''y'' and each ''x''-edge to the similarly ordered ''y''-edge. This property is shared only by the diamond crystal (the strong isotropy should not be confused with the edge-transitivity or the notion of

symmetric graph In the mathematical field of graph theory, a graph is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices and of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 and f(v_1) = v_2. In oth ...

; for instance, the primitive cubic lattice is a symmetric graph, but not strongly isotropic). The K₄ crystal and the diamond crystal as networks in space are examples of “standard realizations”, the notion introduced by Sunada and Motoko Kotani as a graph-theoretic version of Albanese maps ( Abel-Jacobi maps) in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

. For his work, see also

Isospectral In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity. The theory of isospectra ...

Reinhardt domain The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...

Ramanujan graph In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. AMurty's survey papernotes, Ramanu ...

quantum ergodicity In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the Quantization (physics), quantization of classical mechanics, classical mechanical systems that are chaos theory, chaotic in the sense of exponential sensit ...

quantum walk Quantum walks are quantum analogues of classical random walks. In contrast to the classical random walk, where the walker occupies definite states and the randomness arises due to stochastic transitions between states, in quantum walks randomness ...

Selected publications by Sunada

* T. Sunada, Holomorphic equivalence problem for bounded Reinhardt domains, ''

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

'' 235 (1978), 111–128 * T. Sunada, Rigidity of certain harmonic mappings, ''

Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...

'' 51 (1979), 297–307 * J. Noguchi and T. Sunada, Finiteness of the family of rational and meromorphic mappings into algebraic varieties, ''

American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United ...

'' 104 (1982), 887–900 * T. Sunada, Riemannian coverings and isospectral manifolds, ''

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...

'' 121 (1985), 169–186 * T. Sunada, ''L''-functions and some applications, ''Lecture Notes in Mathematics'' 1201 (1986), Springer-Verlag, 266–284 * A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemann surface, ''

'' 110(1988), 145–156 * T. Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, ''

Topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

'' 28 (1989), 125–132 * A. Katsuda and T. Sunada, Closed orbits in homology classes, ''

Publications Mathématiques de l'IHÉS ''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche ...

'' 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, ''

Communications in Mathematical Physics ''Communications in Mathematical Physics'' is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, sta ...

'' 209 (2000), 633–670 * M. Kotani and T. Sunada, Spectral geometry of crystal lattices, ''Contemporary Mathematics'' 338 (2003), 271–305 * T. Sunada, Crystals that nature might miss creating, ''Notices of the American Mathematical Society'' 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 Journal of Mathematics'' 7 (2012), 1–39 * T. Sunada, ''Topological Crystallography, With a View Towards Discrete Geometric Analysis'', Springer, 2013, (print) (online) * T. Sunada, Generalized Riemann sums, ''in From Riemann to Differential Geometry and Relativity'', Editors: Lizhen Ji, Athanase Papadopoulos, Sumio Yamada, Springer (2017), 457–479 * T. Sunada, Topics on mathematical crystallography, ''Proceedings of the symposium Groups, graphs and random walks'', London Mathematical Society Lecture Note Series 436, Cambridge University Press, 2017, 473–513 * T. Sunada, From Euclid to Riemann and beyond, ''in Geometry in History'', Editors: S. G. Dani, Athanase Papadopoulos, Springer (2019), 213–304

References

*Atsushi Katsuda and Polly Wee S

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 (200

* Diamond twin, Meiji U. Homepag

{{DEFAULTSORT:Sunada, Toshikazu 1948 births Living people 20th-century Japanese mathematicians 21st-century Japanese mathematicians Geometers Meiji University faculty Tokyo Institute of Technology alumni Tohoku University faculty Nagoya University faculty