Title: The spacing statistics of Farey fractions and the horocycle flow in $SL(2,\mathbb{R})$
Speaker: Byron Heersink (The Ohio State University)
Abstract: Work of Athreya and Cheung connected Farey fraction statistics to dynamics by constructing a 2 dimensional subspace of $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$, which is a cross section transversal to the horocycle flow in such a way that the Farey fractions are in a sense encoded by the intersections of certain horocycles and the section. In this talk, we discuss how to lift the section to the larger space $SL(2,\mathbb{R})/H$, where $H$ is a finite index subgroup of $SL(2,\mathbb{Z})$, that we utilize to establish statistical and Diophantine results for various subsets of Farey fractions. In the case where H is a congruence subgroup, the subsets of Farey fractions we can consider are determined by congruence conditions on numerators and denominators.
Note: The goal of the Welcome Seminar is to give an opportunity to new postdoctoral fellows and new professors to introduce themselves to their colleagues. The talks are intended to be non-technical and accessible to graduate students as well. Please plan to attend and encourage your colleagues to do so as well. It will be preceded by a colloquium-style reception at 4:00, in the area adjacent to CH240 on the second floor of the Math Building.
