Understanding the size of the Laplacian spectral gap on a closed hyperbolic surface provides a plethora of information about the geometry of the surface. The typical size and fluctuations of the gap for a randomly constructed hyperbolic surface are related to random matrix theory from deep physics conjectures. In this talk, I will discuss recent work with Will Hide (Oxford) and Davide Macera (Bonn) where we demonstrate that typical Weil-Petersson random hyperbolic surfaces have a near optimal spectral gap with an explicit polynomial error rate. Our proof uses a fusion of the trace formula and the recent 'polynomial method' introduced by Chen, Garza-Vargas, Tropp and van Handel for proving strong convergence results.
