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.[-]