Toshikazu Sunada
Encyclopedia
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).
, 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 of the Mathematical Society of Japan
(1987).
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 function
s, 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
.
(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.
Japanese people
The are an ethnic group originating in the Japanese archipelago and are the predominant ethnic group of Japan. Worldwide, approximately 130 million people are of Japanese descent; of these, approximately 127 million are residents of Japan. People of Japanese ancestry who live in other countries...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and author of many books and essays on mathematics and mathematical sciences. He is professor of mathematics at Meiji University
Meiji University
is a private university in Tokyo and Kawasaki, founded in 1881 by three lawyers of the Meiji era, Kishimoto Tatsuo, Miyagi Kōzō, and Yashiro Misao. It is one of the largest and most prestigious Japanese universities in Tokyo, Japan....
, Tokyo
Tokyo
, ; officially , is one of the 47 prefectures of Japan. Tokyo is the capital of Japan, the center of the Greater Tokyo Area, and the largest metropolitan area of Japan. It is the seat of the Japanese government and the Imperial Palace, and the home of the Japanese Imperial Family...
, and is also professor emeritus of Tohoku University
Tohoku University
, abbreviated to , located in the city of Sendai, Miyagi Prefecture in the Tōhoku Region, Japan, is a Japanese national university. It is the third oldest Imperial University in Japan and is a member of the National Seven Universities...
, Tohoku, Japan
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
. Before he joined Meiji University in 2003, he was professor of mathematics at Nagoya University
Nagoya University
Nagoya University is one of the most prestigious universities in Japan. It can be seen in the several rankings such as shown below.-General Rankings:...
(1988–1991), at the University of Tokyo
University of Tokyo
, abbreviated as , is a major research university located in Tokyo, Japan. The University has 10 faculties with a total of around 30,000 students, 2,100 of whom are foreign. Its five campuses are in Hongō, Komaba, Kashiwa, Shirokane and Nakano. It is considered to be the most prestigious university...
(1991–1993), and at Tohoku University (1993–2003).
Main work
Sunada's work covers complex analytic geometryComplex analytic geometry
In mathematics, complex analytic geometry sometimesdenotes the application of complex numbers to plane geometry.Rather than represent a point in the plane as a pair of Cartesian coordinates, it can be represented as a single complex number, which can be written at will in either rectangular or...
, spectral geometry
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 manifold has been most intensively studied, although other...
, dynamical systems, probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, 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
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, and is considered to be a breakthrough in the problem proposed by Mark Kac
Mark Kac
Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D...
in "Can one hear the shape of a drum?" (see Hearing the shape of a drum
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 basic harmonics, via the use of mathematical theory...
). 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 of the Mathematical Society of Japan
Mathematical Society of Japan
The Mathematical Society of Japan was the first academic society in Japan.In 1877, the organization was establisehd as the Tokyo Sugaku Kaisha. It was re-organized and re-established in its present form in 1946....
(1987).
In a joint work with Atsushi Katsuda, Sunada also established a geometric analogue of Dirichlet's theorem on arithmetic progressions
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 ≥ 0. In other words, there are infinitely many primes which are...
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
Ihara zeta function
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 paths to the spectrum of the adjacency matrix. The Ihara zeta-function was first defined by Yasutaka Ihara in the 1960s in the context of discrete...
s, a discrete analogue of periodic magnetic Schroedinger operators as well as the large time asymptotic behaviors of random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
on crystal lattices. The study of random walk led him to the discovery of a "mathematical twin" of the diamond
Diamond
In mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...
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
Reinhardt domain
In mathematics, especially several complex variables, an open subset G of Cn is called Reinhardt domain if \in G implies \in G for all real numbers \theta_1, \dots, \theta_n....
, Ihara zeta function
Ihara zeta function
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 paths to the spectrum of the adjacency matrix. The Ihara zeta-function was first defined by Yasutaka Ihara in the 1960s in the context of discrete...
.
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 LavesFritz Laves
-References:...
(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
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
), "(10,3)-a" (A.Wells), "srs" (M. O'Keeffe), and "triamond" (John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
). The K4 crystal has a close relationship with the gyroid
Gyroid
"Gyroid" redirects here. For the creature, see Animal Crossing .A gyroid is a certain infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970.The gyroid has space group Iad...
, 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
Complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
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
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.
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
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
). This strong isotropic property is the strongest one among all possible meanings of isotropy. In two-dimension, only honeycomb lattice (graphene
Graphene
Graphene is an allotrope of carbon, whose structure is one-atom-thick planar sheets of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice. The term graphene was coined as a combination of graphite and the suffix -ene by Hanns-Peter Boehm, who described single-layer...
) 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