We prove a couple of general conditional convergence results on ergodic averages for horocycle and
geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of push-forwards of horocycle measures under the geodesic flow and a new short proof of a theorem of Chaika and Eskin on Birkhoff genericity in almost all directions for the Teichmüller geodesic flow.[-]

In this talk we explain results on the large genus asymptotics for intersection numbers between \psiclasses on the moduli space of curves. By combining this result with a combinatorial analysis of formulas of Delecroix-Goujard-Zograf-Zorich, we further describe some features about how random flat surfaces of large genus look. The proof uses a comparison between the recursive relations (Virasoro constraints) that uniquely determine them with the jump probabilities of a certain asymmetric simple random walk.[-]

I will present a formula giving the Masur-Veech volumes of 'completed' odd strata of quadratic differentials as a sum over stable graphs. This formula generalizes Delecroix-G-Zograf-Zorich formula in the case of principal strata. The coefficients of the formula are in this case intersection numbers of psi classes with the Witten-Kontsevich combinatorial classes. They naturally appear in the count of integer metrics on ribbon graphs with prescribed odd valencies. The study of the possible degenerations of these ribbon graphs allows to express the difference between the volume of the 'completed' stratum and the volume of the stratum as a linear combination of volumes of boundary strata, with explicit rational coefficients. Several conjectures on the large genus asymptotics of volumes or distribution of cylinders follow from this formula. (work in progress with E. Duryev).[-]

In this talk I will consider the problem of counting the number of pairs (z,w) of saddle connections on a translation surface whose holonomy vectors have bounded virtual area. That is, we fix a positive A and require that the absolute value of the cross product of the holonomy vectors of z and w is bounded by A. One motivation is the result of Smillie-Weiss that for a lattice surface there is a constant A such that if z and w have virtual area bounded by A then they are parallel. We show that for any A there is a constant $c_A$ such that for almost every translation surface the number of pairs with virtual area bounded by A and of length at most R is asymptotic to $c_AR^2$ as R goes to infinity. This is joint work with Jayadev Athreya and Samantha Fairchild.[-]

A meromorphic differential on a Riemann surface is said to be real-normalized if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. We propose a combinatorial model for the real normalized differentials with a single order 2 pole and use it to analyze certain ergodic properties of the corresponding absolute period foliation. It is a joint work with Igor Krichever and Sergey Lando.[-]