Title: Slope gap distributions of translation surfaces
Speaker: Taylor McAdam, Department of Mathematics, Pomona College
Abstract: How “random” are the rational numbers? To make sense of this question, let us consider the set of Farey fractions of level n—that is, the rational numbers between 0 and 1 with denominator at most n. It turns out that these distribute uniformly in the unit interval as n goes to infinity, which would suggest they appear to be quite random. However, we may consider a finer test of randomness by considering the distribution of gaps between consecutive Farey fractions as n tends to infinity. To investigate this, we will first realize the Farey fractions as the slopes of geodesic paths on the (square) flat torus—a geometric object obtained by gluing the opposite edges of a square together. We will then define the horocycle flow on the space of all flat tori, which will allow us to study our question about gaps between Farey fractions via a dynamical system. Finally, we will see how this method can be generalized to study the slope gap distributions for paths on a larger class of geometric objects called translation surfaces and discuss results on the collection of surfaces obtained by gluing together opposite edges of the regular 2n-gon.
Taylor McAdam graduated with a Bachelor’s degree in mathematics from Harvey Mudd College in 2013 before starting a doctoral program at University of Texas at Austin. In 2017, she transferred to the University of California San Diego, where she received her PhD in mathematics in 2019 under the supervision of Amir Mohammadi. She was an NSF Postdoctoral Fellow at Yale University from 2019 to 2023, before joining the faculty at Pomona College in 2023 as a Visiting Assistant Professor. Her research interests lie at the intersection of dynamical systems, geometry, and number theory, and she is passionate about undergraduate math education and building inclusive mathematical communities.