Seminar Calendar
for Graduate Student Number Theory events the next 12 months of Sunday, January 1, 2017.

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Tuesday, January 24, 2017

Graduate Student Number Theory Seminar
2:00 pm   in 241 Altgeld Hall,  Tuesday, January 24, 2017
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Submitted by amalik10.
Byron Heersink (UIUC)
Poincaré sections for the horocycle flow in covers of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) and applications to Farey fraction statistics
Abstract: For a given finite index subgroup $H\subseteq$SL(2,$\mathbb{Z}$), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) found by Athreya and Cheung to the finite cover SL(2,$\mathbb{R}$)/$H$ of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$). We then relate the properties of this section to the gaps in Farey fractions and describe how the ergodic properties of the horocycle flow can be used to obtain certain statistical properties of various subsets of Farey fractions.