Title: New bounds for equiangular lines and spherical two-distance sets

Date: 01/31/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

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}, alpha in [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 R^n is n(n+1)/2 with possible exceptions for some n = (2k+1)^2-3, k a positive integer. We also prove the universal upper bound ~ 2 n a^2/3 for equiangular sets with alpha = 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.