Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials

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Date(s) - 29/03/2019
11 h 00 min - 12 h 00 min

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I’ll begin by discussing Selberg’s eigenvalue conjecture, that predicts a uniform spectral gap for the Laplacian on a special family of arithmetic Riemann surfaces.
Selberg’s conjecture can be restated in terms of the Teichmuller dynamics of abelian differentials on a torus. Based on this, Yoccoz made a generalization of Selberg’s conjecture for connected components of strata of abelian differentials on higher genus surfaces.
I will explain how I have proved an approximation to Selberg’s conjecture in higher genus, and highlight some of the interesting ingredients involved, including the recent resolution of a conjecture of Zorich by Avila-Matheus-Yoccoz and Gutierrez-Romo. If I have time, I’ll explain how this all fits into a broader program of automorphic forms on moduli spaces.

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