Study of generating functions viewed from group theory and category theory

1. Asai-Yoshida关于大量群体同源性的概念与p-adic anallysis有联系（Yoshida、Takegahara、K. Conrad等）。Furthermore， Yoshida解决了一个基本群体中紧凑型Riemann表面对量子场理论的重要性的概念，M.Mulase.2.Yoshida和Oda总结了一个交叉的燃烧边线的有限群体的基本理论，3.Yoshida， Sasaki， Oda研究的有限群体同源理论，特别是Hochschild同源学，交叉的Mackey函数和量子双重代数。Bannai和Keisuke Shiromoto满足组合学，在特定、距离正则图形、设计和代码理论中，特别是将同源算法应用于一个环上的代码理论.5.Koshitani研究对有限组的模块化表示，并给一些特殊defect组的Broue概念提供了一个友好的答案.6.Nakamura和Yamashita研究了一些相关领域（代数几何学和表示理论）和一些有趣的结果，特别是与少数几个简单群体的关系。7.我们邀请了来自国外的三位数学家。2001 FAN Yun（Wuhan U.）《谈论Broue概念》（九州大学）2003年Keith Conrad（康涅狄格州）《P-adic分析：数字理论的谈话》（北海道大学）Kyoto U.）2004年Segre BOUC（CNRS） Talk on Dade Groups and Burnside Rings （Hokaido U.）京都U.）本研究的结果的数目很大，已被印刷和出版。剩余的结果将被正式公布。“Generating functions and related topics”（Sapporo 2001）、“20-th Symposium on Algebraic Combinatorics”（Kyoto 2002）、“Extended Group Seminar”（Sapporo 2002、2004、and so on）。

1.Asai-Yoshida's conjecture on the number of group homomorphisms has a connection with p-adic anallysis (Yoshida, Takegahara, K.Conrad, etc.). Furthermore, Yoshida solved the conjecture for the homomorphisms from the fundamental group of a compact Riemann surface whose importence in quantum field theory was pointed by M.Mulase.2.Yoshida and Oda accomplished a fundamental theory of crossed Burnside rings of finite groups.3.Yoshida, Sasaki, Oda study cohomology theory of finite groups, especially Hochschild cohomology, crossed Mackey functors and quantum doubles of group algebra.4.Yoshida, Bannai and Keisuke Shiromoto sutudies combinatorics, in particular, distance regular graphs, designs and code theory, especially an application of homological algebra to the theory of codes on a ring.5.Koshitani studies modular representation thoery of finite groups and gave an affirmative answer to the Broue conjecture for groups with some special defect groups.6.Nakamura and Yamashita studied some related areas (algebraic geometry and representation theory) and gave some interested results, especially a relation with finite simple groups.7.We invited three mathematicians from abroad.・2001 FAN Yun(Wuhan U.) Talk on Broue conjecture(Kyushu Univ.)・2003 Keith CONRAD(Connecticut U.) Talk of p-adic analysis, number theory(Hokkaido U., Kyoto U.)・2004 Segre BOUC(CNRS) Talk on Dade groups and Burnside rings (Hokaido U., Kyoto U.)A large number of results of this research has been printed and published. The remainder results will be serially published. The investigator gave some talks related to this research in some conferences--"Generating functions and related topics" (Sapporo 2001), "20-th Symposium on Algebraic Combinatorics (Kyoto 2002), "Extended Group Seminar" (Sapporo 2002,2004, and so on.