Geometry of random hyperbolic and flat surfaces of large complexity

Consider a random hyperbolic surface of large genus (where "random" is understood in terms of the Weil-Petersson volume on the moduli space). We know its curvature and its area, but what can be said about its diameter, systole, Cheeger constant, first nonzero eigenvalue of the Laplacian?

Following pioneering works of Maryam Mirzakhani a lot of progress was obtained in recent years in unveiling geometry of random hyperbolic surfaces of large genus or of fixed genus but with many cusps. After a presentation of selected recent results of my colleagues concerning hyperbolic geometry, I will announce the first results on geometry of random flat surfaces where "random" is understood in terms of the Masur--Veech volume on the moduli space of quadratic differentials in large genus or in genus zero but with large number of poles.

 

Based on joint work in progress with Giovanni Forni and Kasra Rafi.