Department of

Mathematics


Seminar Calendar
for Graduate Student Number Theory Seminar events the year of Thursday, September 14, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 24, 2017

2:00 pm in 241 Altgeld Hall,Tuesday, January 24, 2017

Poincaré sections for the horocycle flow in covers of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) and applications to Farey fraction statistics

Byron Heersink (UIUC)

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.

Thursday, September 14, 2017

2:00 pm in 241 Altgeld Hall,Thursday, September 14, 2017

On the percentage of critical zeros of Riemann's zeta function

Kyle Pratt   [email] (UIUC)

Abstract: The Riemann hypothesis (RH) is one of the most important unsolved problems in number theory. RH asserts that all of the important zeros of the Riemann zeta function lie on a specific line, called the critical line. As we lack a solution to RH, it is natural to ask for partial results instead. One way to measure progress towards RH is to prove that some percentage of the zeros are on the critical line. I will sketch a brief history of the results about percentages of zeros on the critical line, and discuss some of the methods of proof. In the latter part of the talk I will discuss the current world record, due to Nicolas Robles and myself, and some of our ideas. The talk should be accessible to any graduate student.