Equiangular lines, Hilbert's 12th problem, and a bit of quantum mechanics

How many lines can one find in high-dimensional space, every pair of which meets at the same angle? How large a multiplicity can the second-largest eigenvalue of a sparse graph have? How can one generate abelian extensions of a real quadratic field? What are the "nicest" generalized measurements in quantum mechanics? In this talk, I will explain the (partly conjectural) relationships between these questions, and describe some of what is known about them.