Mathematical Sciences: Algebraic Combinatorics: Association Schemes and Related Topics

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This work is in the area of algebraic combinatorics. It includes association schemes ((P andQ)-polynomial association schemes and more general association schemes), distance-regular graphs, tight designs, perfect codes and spherical designs. Emphasis will be put on the interplay of algebraic combinatorics with the theory of orthogonal polynomials and with the representation theory of groups. It will start the general classification problem for general commutative association schemes of large class number. More precisely, it will collect many examples and calculate their character tables and find as many necessary conditions for the character table as needed for the classification. It will also study distance-regular graphs with fixed (small) valency and generalize such results to general commutative association schemes with small valency. It will also study extremal configurations of (finite) sets in association schemes and also in the sphere and other (nice) metric spaces. This research will concentrate on combinatorial aspect of algebraic problems. The combination of algebraic, number theoretic and analytic tools necessary to perform this research makes the subject very attractive. Bannai is one of the top leaders in this subject. Result of very great importance will surely result from this project.