Comprehensive study on Algebraic Combinatorics

这笔赠款的目标之一是通过研讨会和旅行援助，广泛促进代数组合学在日本的发展。在2001-2003财政年度期间，这笔赠款用于部分支助：第18、19、20届代数组合专题讨论会(分别为(千叶大学、熊本大学、北洋大学)，在RIMS举办了三次题为“代数组合学”的研讨会，在滨松举办了三次关于自构形相关领域的讲习班(井山、班奈、斋藤、宫友组织者)。山形研讨会(2002年11月)，在福冈举行的EA-CAC2国际会议(Bannai组织者，2003年11月)。日本在代数组合学方面的研究正在稳步发展。这一进展涵盖了距离正则图、结合方案、码、设计、格、模形式等多个领域。主要发明者的结果包括：(I)交换结合方案和Ramanujan图的特征标表，(Ii)码、格和模形式之间的关系，(Iii)球面上的紧4-、5-、7-设计(Bannai Munemasa-Venkov)，(Iv)Grassmanian空间上的紧设计/码(Bacho-Bannal Coulangeon)。此外，价格调查员研究了欧氏空间上的设计：与Etsuko Bannai合作，完成了等权紧4-设计的分类，高斯紧4-设计的分类，以及两个同心球面上的最优紧4-设计的分类。我们目前正在挑战更一般的分类。我们与田部诚共同参与了对奥列格·穆辛壮观结果的验证：确定了四维中的接吻数，并帮助完成了穆辛的证明。

One of the goal of this grant is to widely contribute to the advancement of algebraic combinatorics in Japan through workshops and travel aid. During the fiscal year 2001-2003, the grant was used to partly support : The 18th, 19th, 20th Algebraic Combinatoric Symposiums (at resp. Chiba University, Kumamoto University, Hokkaio University), three symposiums entitled "Algebraic Combinatorics" at RIMS, three workshops on areas related to automorphic forms at Hamamatsu (Ibukiyama, Bannai, Saito, Miyamoyo organizers). Symposium at Yamagata (Nov. 2002), International Conference EA- CAC2 held in Fukuoka (Bannai organizer, Nov. 2003).Research in Algebraic Combinatorics in Japan is showing steady progress. The progress covers diverse areas such as distant-regular graphs, association schemes, codes, designs, lattices, modular forms. The results of the principal invstigator includes: (i) Character tables of commutative association schemes and Ramanujan graphs, (ii) Relations between codes, lattices and modular forms, (iii) Tight 4-, 5-, 7-designs on spheres (Bannai Munemasa-Venkov), (iv) Tight designs/codes over Grassmanian spaces (Bachoo-Bannal Coulangeon). In addition, the pricinpal investigator studied designs on Euclidean spaces: Collaborating with Etsuko Bannai, the classification of tight 4-designs with constant weight, the classification of Gaussian tight 4-designs, and the classification of optimal tight 4-designs on two concentric spheres were completed. We are currently challenging more general classi-fications. Jointly with Makoto Tanabe, we participated in the verification of Oleg Musin's spectacular result: The determination of the Kissing number in dimension four, and helped the completion of Musin's proof.