Abstract

The problem of equiangular lines asks how many lines through the origin can be placed in the Euclidean space while maintaining the pairwise angle of lines to be constant. This problem of metric geometry is seemingly easier than packing problems in the sphere or the Euclidean spaces, but it is in general open except for some small dimensions. By Lemmens and Seidel (1973), the problem can be reformulated as a purely combinatorial problem in terms of switching classes of graphs. In this talk, I will survey recent development on this problem in connection with root systems and integral lattices. I will also briefly touch on its complex analogue, where ``lines'' mean complex 1-dimensional subspaces, and ``angles'' mean the absolute values of the inner product of representing unit vectors. This analogue has been investigated by quantum physicists under the name SIC-POVM, yet its existence problem is unsolved in general.

Biography

Akihiro Munemasa received the B.Sci. and M.Sci. degrees from Sophia University, located in Tokyo, Japan, in 1983 and 1985, respectively. In 1985, he became a student of the Ohio State University, Columbus, Ohio, U.S.A., and earned a Ph.D. degree in 1989 under the supervision of Professor Eiichi Bannai. He held research/teaching positions at Osaka Kyoiku University, Osaka, Japan and at Kyushu University, Fukuoka, Japan, before he got promoted to a full professor at Tohoku University, Sendai, Japan in 2003. He was one of the four Editors-in-Chief of Journal of Algebraic Combinatorics from 2000 to 2017, and then joined the team of Editors-in-Chief of the new open-access journal Algebraic Combinatorics in 2018. His research interest includes algebraic coding theory, association schemes, combinatorial design theory, spectral graph theory, number theory and permutation groups. He has published more than 100 refereed research articles in these areas, and delivered a number of invited talks at international conferences. He also gave series of lectures in international workshops held in Russia, Canada and Thailand. He produced 13 Ph.D's and, mentored 3 Japanese and 3 international postdoctoral fellows.