Colloque - Géométries aléatoires et applications - Yilin Wang : The Brownian Loop Measure on Riemann Surfaces and Applications to Length Spectra

Nalini Anantharaman

Géométrie spectrale

Collège de France

Année 2024-2025

Colloque - Géométries aléatoires et applications - Yilin Wang : The Brownian Loop Measure on Riemann Surfaces and Applications to Length Spectra

Intervenant :

Yilin Wang

IHES

Résumé

The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. More precisely, we use the Brownian loop measure to express the lengths of closed geodesics on a hyperbolic surface and zeta-regularized determinant of the Laplace-Beltrami operator. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a compact surface and that of the same surface with an arbitrary number of additional cusps.

This is a joint work with Yuhao Xue (IHES).