Departmental Colloquium

A family of lines passing through the origin in an inner product space is said to be equiangular if every pair of lines defines the same angle. In 1973, Lemmens and Seidel raised what has since become a central question in the study of equiangular lines in Euclidean spaces. They asked for the maximum number of equiangular lines in R^r with a common angle of alpha. This classical question stems its origins from elliptic geometry, and has since found connections and applications to a large number of different areas including frame theory, Banach spaces, spectral graph theory, algebraic number theory, quantum information theory, and theoretical computer science.

Improving on a number of recent breakthroughs we determine the answer up to lower order terms for essentially the whole range of parameters and determine it precisely when alpha=arccos(1/(2k-1)) for any positive integer k, when the dimension is at least exponential in a polynomial in k. The key new ingredient underlying our results is an improved upper bound on the multiplicity of the second-largest eigenvalue of a graph.