Information, Geometry, and Physics Seminar

Associativity of point-wise multiplication between functions provides constraints on the spectral data of a manifold. In this talk, I will discuss how the method of semi-definite programming allows one to obtain upper bounds for the lowest non-zero eigenvalues of Laplacian and Dirac operator on 2d hyperbolic surfaces and orbifolds equipped with a spin structure. A numerical algorithm based on Selberg trace formula shows the [0;3,3,5] hyperbolic triangle and the Bolza surface nearly saturates the numerical bound at genus 0 and genus 2. This method also produces bounds that are specific to hyperelliptic surfaces. Our approach is inspired by the method of conformal bootstrap, which is a widely adopted technique to extract constraints on conformal field theories from basic self-consistency conditions.