Comprehensive study on Algebraic Combinatorics

该补助金的目标之一是通过讲习班和旅行补助，为日本代数组合学的发展做出广泛贡献。在2001-2003财政年度，该赠款用于部分支持：第18、19、20届代数组合学研讨会（分别在2003年和2004年）。千叶大学、熊本大学、北海道大学），在RIMS举办了三次题为“代数组合学”的专题讨论会，在滨松举办了三次关于自守形式的讲习班（Ibukiyama、Bannai、Saito、Miyamoyo组织者）。2002年11月在山形举行的学术讨论会，2003年11月在福冈举行的国际会议EA-CAC 2（Bannai主办方）。这些进展涵盖了不同的领域，如距离正则图，关联方案，代码，设计，格，模形式。主要研究结果包括：（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（Bachoo-Bannal Coulangeon）.此外，pricinpal研究者研究了欧几里得空间上的设计：与Etsuko Bannai合作，完成了具有常数权重的紧4-设计的分类，高斯紧4-设计的分类，以及两个同心球上的最佳紧4-设计的分类。我们目前正在挑战更一般的分类。我们与田边诚共同参与了Oleg Musin的惊人结果的验证：确定四维中的Kissing数，并帮助完成了Musin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.