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Thursday, October 23, 2014

**Abstract:** This talk will first discuss how to lift a cross section of the horocycle flow on SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) found by Athreya and Cheung to finite covers SL(2,$\mathbb{R}$)/$H$, $H$ a finite index subgroup of SL(2,$\mathbb{Z}$). As an application, we will establish the limiting gap distribution of various subsets of Farey fractions using the ergodic properties of the horocycle flow. We will then discuss the work of Kessebohmer and Stratmann in applying infinite ergodic theory to the distribution of the Stern-Brocot sequence, and give new related results using more elementary methods.