Explore a seminar talk on the spectral gap of the Laplacian for random hyperbolic surfaces. Delve into the Weil-Petersson probability measure on the moduli space of hyperbolic surfaces and its significance in choosing compact hyperbolic surfaces randomly. Examine the work of M. Mirzakhani and its impact on studying this probabilistic model. Investigate the spectral gap λ1 of the Laplacian for random compact hyperbolic surfaces in the limit of large genus, as presented by Nalini Anantharaman from the Collège de France. Learn about the joint work with Laura Monk, which demonstrates that asymptotically almost surely, λ1 is greater than 1/4−ϵ for any ϵ greater than 0. Understand the proof methodology, including the trace method, asymptotic expansions in powers of g−1 for volume functions, and the "Friedman-Ramanujan property" introduced by J. Friedman in his proof of the Alon conjecture for random regular graphs.