# Géométrie et dynamique dans les espaces de modules

Séminaire Mensuel

## Séminaire à venir - Mai

Le séminaire aura lieu un mercredi par mois de 14h à 15h à l'Institut Henri Poincaré à Paris.  Pour télécharger l'affiche du mois: mai.pdf.

• 15/05/2019 - Gaëtan Borot (Max Planck Institute for Mathematics)  -  salle 01

• Titre: Masur-Veech volumes from topological recursion

Resumé: Statistics of the simple length spectrum of bordered hyperbolic surfaces define functions on the moduli space. Andersen, Orantin and the speaker showed recently that they satisfy a recursion on the Euler characteristic, which implies a topological recursion for their averages over the Weil-Petersson measure. This can be seen as a generalization of Mirzakhani's identity and her proof of a topological recursion for the Weil-Petersson voulmes. We show how this result implies topological recursion (here taking the form of Virasoro constraints) for the Weil-Petersson averages of the asymptotic growth of the number of long curves. By invoking the relation between Weil-Petersson measure on the Teichmuller space, Thurston measure on the space of measured laminations, and Masur-Veech measure on the space of quadratic differentials, this gives a recursion to compute polynomials $$P_{g,n}(L_1,...,L_n)$$ whose constant term are the Masur-Veech volumes. This retrieve and generalizes a result of Delecroix et al. obtained via different (combinatorial) methods. If time permits, I will present some conjectural formulas for Masur-Veech volumes and area Siegel-Veech constants in low genus for any number of punctures. This is based on ongoing joint work with Severin Charbonnier, Vincent Delecroix, Alessandro Giacchetto, Danilo Lewanski and Campbell Wheeler.