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Explore the fascinating world of random hyperbolic surfaces in this 59-minute lecture by Nalini Anantharaman at the Institut Henri Poincaré. Delve into the Weil-Petersson probability measure on the moduli space of hyperbolic surfaces, considered the most natural way to "choose a compact hyperbolic surface at random." Discover how M. Mirzakhani's work has enabled the study of this probabilistic model, one of the few "random Riemannian manifolds" models where explicit calculations are possible. Investigate questions about the geometry and spectral statistics of the Laplacian on randomly chosen surfaces, drawing parallels with inquiries typically made for random graph models. Focus on the spectral gap of the Laplacian for random compact hyperbolic surfaces in the limit of large genus, a topic explored in collaboration with Laura Monk. Gain insights into this unique area of mathematical research that bridges geometry, probability, and spectral theory.
