The Polynomial Method for Spectral Gaps of Random Hyperbolic Surfaces

Explore the polynomial method for analyzing spectral gaps of random hyperbolic surfaces in this 42-minute conference talk. Delve into how the size of the Laplacian spectral gap on closed hyperbolic surfaces reveals crucial geometric information, with typical gap sizes and fluctuations for randomly constructed surfaces connecting to random matrix theory through deep physics conjectures. Discover recent collaborative research demonstrating that typical Weil-Petersson random hyperbolic surfaces exhibit near-optimal spectral gaps with explicit polynomial error rates. Learn how the proof combines the trace formula with the innovative 'polynomial method' developed by Chen, Garza-Vargas, Tropp and van Handel for establishing strong convergence results. Gain insights into the intersection of probability theory, geometry, and spectral analysis through this advanced mathematical presentation delivered at IPAM's New Interactions Between Probability and Geometry Workshop.