Geometry and Topology Seminar: Slope gap distributions: from Farey fractions to translation surfaces

Abstract: The Farey fractions of order n are an ordered list of reduced fractions between 0 and 1 with denominators at most n. For example, F_4 = {0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1}. A natural question that one could ask is: how random are these sets of fractions? One way to measure their randomness is to ask for a function called their limiting gap distribution. While the gap distribution of the Farey fractions has been known for some time, recent work of Athreya and Cheung rederived the gap distribution using dynamical techniques relating to dynamics on moduli spaces of translation surfaces. Translation surfaces are geometric surfaces that are locally Euclidean except at finitely many cone points. In this talk, we will explain what the slope gap distribution of a translation surface is and survey some known results about slope gap distributions, including how one can use dynamical properties of the horocycle flow to compute the slope gap distributions of special translation surfaces called Veech surfaces (the Farey fractions are related to the square torus, an example of a Veech surface). We'll then discuss results showing that the slope gap distributions of Veech surfaces have to satisfy some nice properties. This project is joint work with Luis Kumanduri and Anthony Sanchez.

Host: Charles Ouyang