Conformal bootstrap for hyperbolic surfaces and subconvexity

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 automorphic forms. These numbers obey algebraic constraints analogous to the conformal bootstrap equations in physics. In this talk I will present two results. The first is a converse theorem for these constraints: any collection of numbers satisfying the constraints must come from a hyperbolic surface. The second is an application of the constraints to subconvexity for triple product L-functions. This second result is joint with James Bonifacio, Petr Kravchuk, Dalimil Mazáč, Sridip Pal, Alex Radcliffe, and Gordon Rogelberg. No knowledge of physics will be assumed.