Joint IAS/PU Analysis Seminar

Given a compact hyperbolic surface of fixed topology, we consider its Laplace eigenvalues together with the structure constants for multiplication with respect to a suitable orthonormal basis of Laplace eigenforms. These numbers obey algebraic constraints analogous to the conformal bootstrap equations in physics. The main result of this talk is a converse theorem for these constraints: any collection of numbers satisfying the constraints must come from a hyperbolic surface. 

I will also briefly mention applications of these constraints to upper bounds for spectral gaps and subconvex bounds for L-functions. No knowledge of physics or L-functions will be assumed.