(1)  Algebraic combinatorics

The theory of association schemes has been developed to unify the application of linear programming to coding theory and design theory by Delsarte in 1970's. It generalizes the action of finite groups, and gives a framework for algebraic graph theory, algebraic coding theory and combinatorial design theory. In order to develop algebraic tools for these theories, we investigate applications of algebraic methods to combinatorics, mainly from graph spectra, finite groups, representation theory, linear algebra and optimization.

(2) Codes and lattices

A code is a subspace of a finite-dimensional vector space over a finite field. This seemingly simple concept has been widely used to study combinatorial problems using algebraic methods. Codes themselves can also be investigated from tools in number theory, modular forms in particular, via integral lattices. The class of self-dual codes is an interesting class of codes which give rise to unimodular lattices, and are related to the sphere packing problem and the theory of spherical designs. Moreover, some vertex operator algebras are constructed from codes and lattices. We investigate the problems of construction and classification of codes and lattices, and study their relations.

Research on knots and 3-dim manifolds

I study knots and 3-dim manifolds,and my research topics include quantum invariants and
Heegaard Floer homology. From a representation of a quantum group we can create an R-
matrix, which is a solution to the Yang-Baxter equation. An R-matrix gives a matrix
representation of braid group, and a quantum invariant is constructed in this way. Heegaard
Floer homology is an invariant defined using techniques in symplectic geometry. I am
interested in the topological applications and interpretations of these invariants.