Description:

There was a lot of recent progress in the study of hyperbolic surfaces (and of the associated moduli spaces of complex curves) and in the study of flat surfaces (and of the associated moduli spaces of Abelian and quadratic differentials). Fifteen years ago Maryam Mirzakhani launched a program aimed to answer the question "How does a random hyperbolic surface looks like?" This program had spectacular development in the last years. We will present some of them in parallel in the hyperbolic and in the flat worlds.

Namely, we will study the asymptotic geometry of compact surfaces as the genus tends to infinity, in both hyperbolic and flat settings. In the hyperbolic setting we will use the Weil-Petersson measure and in the flat setting we will use the Masur-Veech measure to model a typical large-genus surface. We will track local statistics of geometric quantities such as diameter, systole, and Cheeger constant. The hyperbolic part follows Mirzakhani’s program, beginning with her recursive computation of Weil-Petersson volumes, including Mirzakhani's count of simple closed geodesics and leading to the Mirzakhani-Petri results on the statistics of short closed geodesics on random large-genus hyperbolic surfaces.

On the flat side we will introduce Masur-Veech volumes and we will count flat closed geodesics. We will discuss an average count given by the Siegel-Veech formula. On the way we will present necessary facts about Teichmuller geodesic flow and about the Magic Wand Theorem proved by Eskin and Mirzakhani. We aim to arrive to two recent results on the flat side. First, we will obtain quantitative control on the lengths of saddle connections in large genus, including the typical scale at which short saddle connections appear and the expected number of such connections. This will provide a flat-surface analogue of the Mirzakhani-Petri results for short geodesics. Throughout we emphasize the shared toolkit and the contrasts between the two settings. Finally, if time allows, we plan to describe the Benjamini-Schramm limit of translation surfaces: as the genus grows, we identify the local limiting geometry around a random point and show that the limit is a simply connected translation surface with cone points distributed by a Poisson point process. This parallels the hyperbolic picture, where random large-genus hyperbolic surfaces converge locally to the hyperbolic plane.

We do not require any prerequisites, we will define most of the objects on the way. The course will be very informal and would contain only ideas of proofs. This course has close relations in certain aspects with the courses of Yitwah Cheung, Vladimir Markovic and of Don Zagier.

Kasra Rafi: https://www.math.toronto.edu/~rafi/

Anton Zorich: https://webusers.imj-prg.fr/~anton.zorich/ 

Registration: https://www.wjx.top/vm/efW3stb.aspx#