Counting social interactions for discrete subsets of the plane

Given a discrete subset V in the plane, how many points would you expect there to be in a ball of radius 100? What if the radius is 10,000? Due to the results of Fairchild and forthcoming work with Burrin, when V arises as orbits of non-uniform lattice subgroups of SL(2,R), we can understand asymptotic growth rate with error terms of the number of points in V for a broad family of sets. A crucial aspect of these arguments and similar arguments is understanding how to count pairs of saddle connections with certain properties determining the interactions between them, like having a fixed determinant or having another point in V nearby. We will spend the first 40 minutes discussing how these sets arise and counting results arise from the study of concrete translation surfaces. The following 40 minutes will be spent highlighting the proof strategy used to obtain these results, and advertising the generality and strength of this argument that arises from the computation of all higher moments of the Siegel--Veech transform over quotients of SL(2,R) by non-uniform lattices.