### Geometry and spectrum of random hyperbolic surfaces

Analysis and Geometry Seminar

24th March 2022, 3:15 pm – 4:15 pm

Fry Building, Room 2.04

The aim of this talk is to present recent progress in the spectral theory of hyperbolic surfaces, which was achieved by bringing a new probabilistic viewpoint to this classic field. Studying random hyperbolic surfaces allows to prove statements true for most surfaces rather than every single one of them, and hence to exclude some known pathological examples. I will present estimates on the density of eigenvalues and the spectral gap of typical surfaces in the Weil--Petersson random setting. These results are proven using the Selberg trace formula, together with geometric information on closed geodesics on random surfaces.

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