Pairs of saddle connections on translation surfaces
In 2022, Athreya-Fairchild-Masur proved that, for a generic translation surface, the number of pairs of saddle connections of length at most and cross product at most $A$ grows asymptotically like $c_AR^2$ for some (inexplicit) constant $c_A$. In this talk, I will discuss some further work on understanding these asymptotics. This includes uniform bounds and growth rate of $c_A$ as a function of $A$, pointwise results, and a family of exceptional surfaces for which quadratic growth of the associated counting function fails. I will also discuss pointwise convergence for Veech curves, as well as families of non-Veech rank-1 subvarieties. This is partially joint work with Samantha Fairchild, Kanishka Katipearachchi, and Howard Masur.

