Comprehensive study on Algebraic Combinatorics

该基金的目标之一是通过研讨会和旅行援助，为日本代数组合学的进步做出广泛贡献。在2001-2003财政年度，这笔拨款部分用于支持：第18届、第19届、第20届代数组合学术研讨会(分别为：千叶大学、熊本大学、北海洋大学)，在RIMS举办了三场题为“代数组合学”的研讨会，在滨松举办了三场与自同构形式相关的研讨会（Ibukiyama, Bannai, Saito， Miyamoyo组织者）。山形研讨会（2002年11月），国际会议EA- CAC2在福冈举行（班奈组织者，2003年11月）。日本的代数组合学研究正在稳步发展。这一进展涵盖了不同的领域，如远正则图、关联方案、编码、设计、格、模形式。主要研究结果包括：(i)交换关联方案和Ramanujan图的特征表，（ii）码、格和模形式之间的关系，（iii）球上的紧4-、5-、7-设计（Bannai Munemasa-Venkov）， （iv） Grassmanian空间上的紧设计/码（bachooo - bannal Coulangeon）。此外，主要研究欧几里得空间上的设计：与Etsuko Bannai合作，完成了定权紧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Makoto Tanabe共同参与了Oleg Musin的惊人成果的验证：四维接吻数的确定，并帮助完成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.