Information, Geometry, and Physics Seminar

Given a multiset S of nonnegative real numbers and an integer g at least 2, when is S the Laplace spectrum of a closed hyperbolic surface of genus g? Recently, it was observed by physicists that one can associate to each pair (S,g) an infinite system of bilinear equations in infinitely many variables which has a solution if S is such a Laplace spectrum. They then used these equations to prove nearly sharp bounds on low energy eigenvalues of low genus surfaces. In this talk, as a step toward explaining why their bounds are so sharp, I will describe a converse theorem: if the equations admit a solution, and if S obeys a weak form of the Weyl law, then S is such a Laplace spectrum. There is a strong analogy between this story and the conformal bootstrap approach to conformal field theory. From the bootstrap point of view, a CFT is specified by its spectrum of scaling dimensions together with its OPE coefficients, where the OPE coefficients satisfy an infinite system of bilinear "crossing equations." The ultimate goal of the bootstrap is to classify all possible spectra and OPE coefficients which satisfy the crossing equations. My main theorem can be regarded as realizing this goal in the analogous setting of hyperbolic surfaces.