Department of

Mathematics


Seminar Calendar
for Graduate Student Number Theory Seminar events the year of Sunday, May 19, 2019.

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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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                                               30                  

Thursday, April 11, 2019

2:00 pm in 241 Altgeld Hall,Thursday, April 11, 2019

Conversations on the exceptional character

Abstract: We will spend the last few weeks of the semester discussing Landau-Siegel zeros. In particular, we will be discussing Henryk Iwaniec's survey article "Conversations on the exceptional character."

Friday, August 30, 2019

3:00 pm in 343 Altgeld Hall,Friday, August 30, 2019

Organizational Meeting

Aubrey Laskowski (UIUC)

Abstract: This will be the organizational meeting for the graduate student number theory seminar. We will discuss the schedule for weekly meetings, as well as begin sign-up for speakers.

Friday, September 6, 2019

3:00 pm in Illini Hall 1,Friday, September 6, 2019

Series and Polytopes

Vivek Kaushik (Illinois Math)

Abstract: Consider the series $S(k)=\sum_{n \geq 0} \frac{(-1)^{nk}}{(2n+1)^k}$ for $k \in \mathbb{N}.$ It is well-known that $S(k)$ is a rational multiple of $\pi^k$ using standard techniques from either Fourier Analysis or Complex Variables. But in this talk, we evaluate $S(k)$ through multiple integration. On one hand, we start with a $k$-dimensional integral that is equal to the series in question. On the other hand, a trigonometric change of variables shows the series is equal to the volume of a convex polytope in $\mathbb{R}^k.$ This volume is proportional to a probability involving certain pairwise sums of $k$ independent uniform random variables on $(0,1).$ We obtain this probability using combinatorial analysis and multiple integration, which ultimately leads to us finding an alternative, novel closed formula of $S(k).$