3rd KUIP Symposium [Mathematical Sciences] Toshikazu Sunada

July 10, 2016
3rd Kyoto University-Inamori Foundation Joint Kyoto Prize Symposium
http://kuip.hq.kyoto-u.ac.jp/
http://ocw.kyoto-u.ac.jp/en/opencourse-en/158

[Mathematical Sciences]
Toshikazu Sunada
Dean, School of Interdisciplinary Mathematical Sciences, Meiji University

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.