Brownian Loop Measure on Riemann Surfaces and Length Spectra

This lecture explores the Brownian loop measure on Riemann surfaces and its connection to length spectra. Discover how the Brownian loop measure, a sigma-finite measure on Brownian-type loops on the Riemann sphere with conformal invariance and restriction properties, extends to arbitrary Riemann surfaces. Learn how the lengths of closed geodesics for constant curvature metrics are encoded in this measure, providing a valuable tool for studying length spectra of Riemann surfaces. The presentation reveals a new identity between the length spectrum of a surface and that of the same surface with additional cusps, while also expressing the determinant of Laplacian of a compact hyperbolic surface as the total mass of Brownian loop measure renormalized according to the length spectrum. This 53-minute talk from the Hausdorff Center for Mathematics is based on joint work with Yuhao Xue (IHES).