Équations différentielles motiviques et au–delà
# Random square-tiled surfaces and random multicurves in large genus

→
Europe/Paris

salle Pierre Grisard (ex salle 314) (IHP, Paris 75005)
### salle Pierre Grisard (ex salle 314)

#### IHP, Paris 75005

Description

(after joint works with V. Delecroix, E. Goujard and P. Zograf)

Moduli spaces of Riemann surfaces and related moduli spaces of

quadratic differentials are parameterized by a genus g of the

surface. Considering all associated hyperbolic (respectively flat)

metrics at once, one observes more and more sophisticated diversity

of geometric properties when genus grows. However, most of metrics,

on the contrary, progressively share certain rules. Here the notion

of “most of” has explicit quantitative meaning, for example, in

terms of the Weil-Petersson measure. I will present some of these

recently discovered large genus universality phenomena.

I will use count of metric ribbon graphs (after Kontsevich and

Norbury) to express Masur-Veech volumes of moduli space of

quadratic differentials through Witten-Kontsevich correlators. Then

I will present Mirzakhani's count of simple closed geodesics on

hyperbolic surfaces. We will proceed with description of random

geodesic multicurves and of random square-tiled surfaces in large

genus. I will conclude with a beautiful universal asymptotic

formula for the Witten-Kontsevich correlators predicted by

Delecroix, Goujard, Zograf and myself and recently proved by

Amol Aggarwal.

Organized by

Vladimir Rubtsov, Ilia Gaiur

Vladimir Rubtsov