Abstract : 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.

Keywords : saddle connections