演講公告

演講主題：New bounds for equiangular lines and spherical two-distance sets
主講人：俞偉亘(Department of Mathematics, Michigan State University)
演講日期：2016年11月8日(星期二) 下午2:00 –3:00
演講地點：(光復校區) 科學一館223室
茶會時間：當天下午1:30 (科學一館205室)
摘要內容：
Abstract
The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products ${alpha, -alpha}$, $alphain[0,1)$, are called equiangular. The problem of determining the maximal size of $s$-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an $s$-distance set in two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $mathbb{R}^n$ is $frac{n(n+1)}2$ with possible exceptions for some $n=(2k+1)^2-3$, $k in mathbb{N}$. We also prove the universal upper bound $sim frac 2 3 n a^2$ for equiangular sets with $alpha=frac 1 a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension. Finally, we classify all equiangular sets reaching this new bound.

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