(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.
Vladimir Rubtsov, Ilia Gaiur
