Geometric Analysis and Topology Seminar

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A converse theorem for hyperbolic surface spectra and the conformal bootstrap

Speaker: Anshul Adve, Princeton University

Location: Warren Weaver Hall 512

Date: Friday, January 30, 2026, 11 a.m.

Synopsis:

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.