Counting social interactions for discrete subsets of the plane

Abstract: 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 focus on a concrete case used to state the theorem and highlight the proof strategy. We will also discuss some ongoing work and ideas which advertise the generality and strength of this argument.

Mathematics

Audience: researchers in the topic

New England Dynamics and Number Theory Seminar