Eigenvalue Multiplicity and Equiangular Lines

Equiangular lines are collections of lines in Euclidean space, all passing through the origin and forming the same angle with one another. Given a fixed angle, a fundamental question is: what is the maximum number of such lines? I will present a solution to this problem, with a key step involving the proof that any connected graph with bounded degree has sublinear multiplicity of its second eigenvalue. I will also discuss recent developments and outline possible directions for future research.