Samantha Fairchild

Pairs in discrete lattice surfaces with applications to Veech surfaces

Given a discrete subgroup in SL(2,R), we consider discrete orbits of this subgroup under the linear action on the Euclidean plane. For example the orbit SL(2,Z) acting on the first basis vector (1,0) gives a subset of the integer lattice. Understanding the distribution of these discrete subgroups is a venerable problem going back to Gauss with the study of the Gauss circle problem and popularized through the study of the geometry of numbers in the mid to late 1900's. We will give a Siegel--Veech type integral formula for averages of pairs of discrete lattice orbits. The applications of this formula will focus on Veech surfaces, which are certain surfaces given by polygons in the plane with opposite sides identified. A classical example of a translation surface is the torus presented as a unit square in the plane with opposite sides identified. We'll present the general theory through focusing on the examples of the torus and my other favorite translation surface: the Golden L. This talk is based on work with Claire Burrin with a dynamics application by Jon Chaika.