Department of

# Mathematics

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

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2017           September 2017          October 2017
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5                   1  2    1  2  3  4  5  6  7
6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
13 14 15 16 17 18 19   10 11 12 13 14 15 16   15 16 17 18 19 20 21
20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
27 28 29 30 31         24 25 26 27 28 29 30   29 30 31



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.

Thursday, October 26, 2017

2:00 pm in 241 Altgeld Hall,Thursday, October 26, 2017

#### Combinatorial Proofs of Identities from Ramanujan’s Lost Notebook

###### Hannah Burson (UIUC)

Abstract: In his lost notebook, Ramanujan stated at least 27 identities related to the Rogers-Fine identity. In this talk, I discuss a group of 6 such identities relating to Roger's false theta functions. We give a new combinatorial interpretation and proof of one identity.

Thursday, November 2, 2017

2:00 pm in 241 Altgeld Hall,Thursday, November 2, 2017

#### RAMANUJAN’S LOST NOTEBOOK: HISTORY AND SURVEY

###### Bruce Berndt   [email] (UIUC)

Abstract: In the spring of 1976, while searching through papers of the late G. N. Watson at Trinity College, Cambridge, George Andrews found a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan, generally regarded as India’s greatest mathematician. In view of the fame of Ramanujan’s earlier notebooks, Andrews naturally called these papers Ramanujan’s “lost notebook.” This work, comprising about 650 results with no proofs, arises from the last year of Ramanujan’s life, and represents some of his deepest work. First, we provide a history of the lost notebook. Second, a general description of the contents of the lost notebook will be provided. Third, the remainder of the lecture will be devoted to a survey of some of the most interesting entries in the lost notebook. These include claims in q-series, theta functions, continued fractions, integrals, partitions, and other infinite series.

Thursday, November 9, 2017

3:00 pm in 241 Altgeld Hall,Thursday, November 9, 2017

#### Reed-Muller Codes Achieve Capacity on Erasure Channels

###### Hsin-Po Wang (UIUC)

Abstract: We will talk about this https://arxiv.org/abs/1601.04689. Reed-Muller Codes generalize Reed–Solomon codes (used on CD/DVD/etc) and Hamming codes (used on RAM/etc). For such a family of codes one may ask whether it asymptotically achieves the capacity in the sense of Shannon's information theory. The answer is yes, on certain channels, and we will go through the ideas in the paper.

Tuesday, November 28, 2017

2:00 pm in 241 Altgeld Hall,Tuesday, November 28, 2017

#### Polynomial methods in number theory

###### Dong Dong   [email] (UIUC)

Abstract: Many recent breakthroughs in number theory, combinatorics, harmonic analysis, and PDE involve "polynomial method". What is a polynomial method? In fact, there are different types of polynomial methods. In this talk, I will introduce one simple but extremely powerful such method (other types will be introduced in subsequential talks). Knowledge of Taylor expansion (or mean value theorem) is required to attend this talk.

2:00 pm in 241 Altgeld Hall,Tuesday, November 28, 2017

#### Polynomial methods in number theory

###### Dong Dong   [email] (UIUC)

Abstract: Many recent breakthroughs in number theory, combinatorics, harmonic analysis, and PDE involve "polynomial method". What is a polynomial method? In fact, there are different types of polynomial methods. In this talk, I will introduce one simple but extremely powerful such method (other types will be introduced in subsequential talks). Knowledge of Taylor expansion (or mean value theorem) is required to attend this talk.

Thursday, November 30, 2017

2:00 pm in 241 Altgeld Hall,Thursday, November 30, 2017

#### Exponential sums and the Linear Independence Conjecture

###### William Banks   [email] (University of Missouri)

Abstract: The Linear Independence Conjecture (LIC) for the Riemann zeta-function asserts that the positive ordinates of the nontrivial zeros of zeta(s) are linearly independent over the rationals. In this talk, I will describe some ongoing joint work with N. Ng, G. Martin and M. Milinovich in which we estimate certain exponential sums in order to obtain evidence for the truth of the LIC.

Tuesday, December 5, 2017

2:00 pm in 241 Altgeld Hall,Tuesday, December 5, 2017

#### Zeros of Eisenstein series

###### Oscar Gonzalez Pagan (UIUC)

Abstract: The zeros of the classical Eisenstein series have been well studied over the years. In 1970 it was shown by F. Rankin and Swinnerton-Dyer that the zeros lie on the boundary of the standard fundamental domain. In this talk we will discuss this result as well as some previous results and related problems. We then study the zeros of the object obtained by applying a differential operator to the classical Eisenstein series.