Outline

My research areas are combinatorics and discrete geometry. I am especially interested in objects and configurations with “nice” properties. Those are often called designs. A typical example is the regular icosahedron in 3-dimensional Euclidean space. You can believe that the configuration of the vertices of the regular icosahedron is somehow “nice”; but it might be not so easy to explain why you could feel so. The aim of my research is to combinatorially, or geometrically answer the question “what is nice”. Please do not hesitate to contact me, if you wonna discuss this kind of topic!

Research Subject

(1) Combinatorial Design

In the mid-19^th century Jakob Steiner posed the existence problem of a certain combinatorial object, called a t-design. From the viewpoint of algebraic combinatorics, this problem can be viewed as the problem of extracting suitable orbits of subsets of a finite set under a finite group. There are many rich results for 2-designs available: Hefter (1891), Bose (1942), Wilson (1975) and so on. But, there is still not enough break-through for general t-designs since Carmichael’s theory and Witt’s theory using multiply transitive groups in the early 20^th and Köhler’s theory using cyclic groups in 1979. With these circumstances in mind, I am trying to extend Köhler’s theory so as to be applicable to arbitrary abelian groups.

(2) Cubature Formula, Spherical Design

A cubature formula is a numerical integration formula that evaluates the integrand at “finitely”many points in the space. The well-known Newton-Cotes formula is a one-dimensional cubature formula. For the efficiency of computation, a cubature formula with few points is desirable. The existence of such formulae has long been studied by researchers in analysis: V. A. Ditkin (1948), S. L. Sobolev (1962). Also, a cubature formula with few points is of combinatorial interest: The points of that formula are equally distributed in the space. I am now studying the existence of cubature formulae with few points as well as the geometric structure or configuration of points, by unifying the theory of orthogonal polynomials and that of distance sets, developed in analysis and combinatorics respectively.

(3) Application

In 1994 Kenichi Kitayama proposed a novel type of optical code-division multiple-access (CDMA), called space CDMA, for parallel transmission of 2-dimensional images through a multicore fiber. In space CDMA an optical orthogonal signature pattern code (OOSPC) is required for high efficiency of communications. Most of the known results experimentally examine the possibility of space CDMA systems, and there are only a few papers that theoretically investigate the characteristic of OOSPC. I am thus trying to construct OOSPC through relationships between OOSPC and combinatorial designs with particular algebraic structure.