Mathematical Sciences: Algebraic Combinatorics: Association Schemes and Related Topics

这项工作是在代数组合学领域进行的。它包括结合方案((P和Q)-多项式结合方案和更一般的结合方案)、距离正则图、紧设计、完备码和球面设计。重点放在代数组合学与正交多项式理论和群表示理论的相互作用上。它将启动大类数一般交换结合方案的一般分类问题。更准确地说，它将收集许多示例并计算它们的字符表，并根据分类需要为字符表找到尽可能多的必要条件。它还将研究具有固定(小)度的距离正则图，并将这些结果推广到一般的小度可交换结合方案。它还将研究结合方案中以及球面和其他(NICE)度量空间中(有限)集的极值配置。这项研究将集中在代数问题的组合方面。进行这项研究所必需的代数、数论和分析工具的结合使这一主题非常吸引人。班奈是这一领域的最高领导人之一。这个项目一定会产生非常重要的结果。

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.