# Computer and Mathematical Sciences

## Mathematical Structures I A01

• Prof. Akihiro Munemasa
• Assoc. Prof. Hiroki Shimakura
KeywordsCombinatorics, Algebraic Graph Theory, Vertex Operator algebras, Finite Group Theory

## Research on mathematical theory with algebraic or discrete approach

### (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, lattices and vertex operator algebras

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, lattices and vertex operator algebras and study their relations.

### (3)  Automorphism groups of vertex operator algebras and the Monster simple group

The Monster simple group is a sporadic simple group, and it is related to number theory and operator algebra theory. We investigate its properties and mysterious phenomena from the view point of automorphism groups of vertex operator algebras. In particular, we focus on algebraic and combinatorial structures related to vertex operator algebras.

• Two conjugacy classes of the symmetric group of degree 6

• Y-presentation of the Bimonster