Abstract: The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. More precisely, I will recall the basic facts about surfaces with constant curvature and Brownian motion on them. Then, 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). 

For further information please contact elisur.magrini@unibocconi.it