Department of

Mathematics


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

Thursday, September 28, 2017

2:00 pm in 241 Altgeld Hall,Thursday, September 28, 2017

Extreme values of zeta and $L$-functions

Siegfred Baluyot (UIUC)

Abstract: One of the most important problems in the theory of the Riemann zeta-function is to determine how large the modulus of zeta can be on the critical line. In this talk, we will first outline the history of this problem. Then we will discuss the versatile `resonance method' of Soundararajan for detecting large values of zeta and other $L$-functions. We then complete our discussion with the recent breakthrough of Bondarenko and Seip that significantly improves the resonance method for zeta by using estimates for greatest common divisor sums.

Thursday, October 5, 2017

2:00 pm in 241 Altgeld Hall,Thursday, October 5, 2017

Ramanujan's life and earlier notebooks

Bruce Berndt   [email] (UIUC)

Abstract: Generally regarded as India's greatest mathematician, Srinivasa Ramanujan was born in the southern Indian town of Kumbakonam on December 22, 1887 and died in Madras at the age of 32 in 1920. Before going to England in 1914 at the invitation of G.~H.~Hardy, Ramanujan recorded most of his mathematical discoveries without proofs in notebooks. The speaker devoted over 20 years to the editing of these notebooks; his goal was to provide proofs for all those claims of Ramanujan for which proofs had not been given in the literature. In this lecture, we give a brief history of Ramanujan's life, a history of the notebooks, a general description of the subjects found in the notebooks, and examples of some of the more interesting formulas found in the notebooks.

Thursday, October 19, 2017

2:00 pm in 241 Altgeld Hall,Thursday, October 19, 2017

Some classical applications of modular forms in number theory

Yifan Yang   [email] (National Chiao Tung University)

Abstract: In this talk, we will give a quick overview of some classical applications of modular forms in number theory, including 1. formulas for the number of representations of an integer as sums of squares, 2. a formula for arithmetic-geometric means, 3. modular forms as solutions of linear ordinary differential equations, 4. modular forms as periods, 5. irrationality of $\zeta(3)$, 6. series representations for $1/\pi$, 7. congruences of the partition function.