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).

Bio:

Yilin Wang is a junior professor and mathematician at the Institut des Hautes Études Scientifiques (IHES) in France, working at the interface of probability theory, complex analysis, and hyperbolic geometry. Her research topics are often motivated by questions in mathematical physics. She studied at the École Normale Supérieure (ENS) in Paris (2011-2015) and completed her PhD (2015-2019) at ETH Zurich under the supervision of Wendelin Werner. Prior to joining IHES, she was at the Massachusetts Institute of Technology (MIT) as a C.L.E. Moore Instructor (2019-2022).

Recording: https://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=The_Brownian_loop_measure_on_Riemann_surfaces_and_applications_to_length_spectra.html