Yilin Wang: The Brownian loop measure on Riemann surfaces, length spectra, and determinants of Laplacians

Location

Oskar Klein auditorium (FR4), Albanova

Schedule

14:15–15:00 Pre-colloquium by Ellen Krusell in FB54.

15:15–16:15 Colloquium lecture by Yilin Wang.
16:15–17:00 SMC social get together with refreshments.

Abstract

The Brownian loop measure on the Riemann sphere was introduced by Lawler and Werner while studying Schramm–Loewner evolution. It is a sigma-finite measure on Brownian-type loops, which satisfies conformal invariance and restriction property. We study the generalization of Brownian loop measure on a Riemannian surface (X,g) and express the total mass of loops in every free homotopy class of closed loops on X as a simple function of the length of the geodesic in the homotopy class for the constant curvature metric conformal to g. This identity provides a new tool for studying Riemann surfaces' length spectra. One of the applications is a new identity between the length spectra of a compact surface and that of the same surface with arbitrary additional cusps. We also show that the log determinant of Laplacian of a compact hyperbolic surface equals the total mass of Brownian loops renormalized according to homotopy classes. This is a joint work with Yuhao Xue (IHES).