Department of

# Mathematics

Seminar Calendar
for Graduate Student Number Theory Seminar events the year of Friday, September 14, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2018           September 2018          October 2018
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                      1       1  2  3  4  5  6
5  6  7  8  9 10 11    2  3  4  5  6  7  8    7  8  9 10 11 12 13
12 13 14 15 16 17 18    9 10 11 12 13 14 15   14 15 16 17 18 19 20
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30


Thursday, February 1, 2018

2:10 pm in 241 Altgeld Hall,Thursday, February 1, 2018

#### Modular Forms and Moduli of Elliptic Curves

###### Ningchuan Zhang (UIUC)

Abstract: In this talk, I’ll explain why modular forms are global sections of the sheaf of invariant differentials over the moduli stack of elliptic curves. In the end, I’ll mention the $q$-expansion principle and integral modular forms. No knowledge of stack is assumed for this talk. Please note this talk will start 10 minutes later than the regular time.

Thursday, February 8, 2018

2:00 pm in 241 Altgeld Hall,Thursday, February 8, 2018

#### Multiplicative functions which are additive on polygonal numbers

###### Byungchan Kim (SeoulTech)

Abstract: Spiro showed that a multiplicative function which is additive on prime numbers should be the identity function. After Spiro's work, there are many variations. One direction is to investigate multiplicative functions which are additive on polygonal numbers. It is known that a multiplicative function which is additive on triangular numbers should be the identity while there are non-identity functions which is multiplicative and is additive on square numbers. In this talk, we investigate multiplicative functions which are additive on several polygonal numbers and present some open questions. This talk is based on joint works with J.-Y. Kim, C.G. Lee and P.-S. Park.

Thursday, February 15, 2018

2:00 pm in 241 Altgeld Hall,Thursday, February 15, 2018

#### p-adic families of modular forms

###### Ravi Donepudi   [email] (UIUC)

Abstract: This talk is an introduction to the theme of p-adic variation in number theory, especially concerning modular forms. We will first give a general overview of the p-adic Galois representation attached to a classical Hecke eigenform. Then we will closely study the example of Eisenstein series which are classically parametrized by integer weights and see how they can be naturally interpolated p-adically to give a family of p-adic modular forms. As a corollary, this yields a very simple construction of p-adic L-functions. Finally we will tie these two themes together to study the structure of p-adic Galois representations arising from modular forms in the larger space of all p-adic Galois representations (of a fixed Galois group), highlighting Gouvêa-Mazur’s construction of the “infinite fern of Galois representations" and Coleman-Mazur’s Eigencurve. Have I mentioned the word “p-adic” yet?

Thursday, February 22, 2018

2:00 pm in 241 Altgeld Hall,Thursday, February 22, 2018

#### Multiples of long period small element continued fractions to short period large elements continued fractions

###### Michael Oyengo (UIUC)

Abstract: We construct a class of rationals and quadratic irrationals having continued fractions whose period has length $n\geq2$, and with "small'' partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large'' partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of these continued fractions and study properties of polynomials arising from these generalizations.

Thursday, March 1, 2018

2:00 pm in 241 Altgeld Hall,Thursday, March 1, 2018

#### Spectral Theory on the Modular Surface

Abstract: Many questions about number fields can be recast as questions regarding Laplace eigenvalues on certain manifolds. In this talk I’ll discuss some ideas and results related to Selberg’s trace formula, how partial results towards Selberg’s $\frac{1}{4}$-conjecture have immediate applications, and why number theorists might care about the analysis of Laplacians to begin with.

Thursday, March 8, 2018

2:00 pm in Altgeld Hall,Thursday, March 8, 2018

#### Cancelled

Thursday, March 15, 2018

2:00 pm in 241 Altgeld Hall,Thursday, March 15, 2018

#### Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels

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

Abstract: We will talk about https://arxiv.org/abs/0807.3917 Polar code is considered one of the best codes in the world (together with LDPC and Turbo code). Following Arikan's paper, we will define polar code from scratch and prove that it achieve capacity. If time permits, we will talk about implementation details; in particular comparing virtual channels in an engineering-friendly way. If time still permits, we will talk about how fast does it achieve capacity.

Thursday, March 29, 2018

2:00 pm in 241 Altgeld Hall,Thursday, March 29, 2018

#### Restriction estimates and their applications in number theory

###### Fernando Xuancheng Shao   [email] (University of Kentucky)

Abstract: I will survey recent developments on restriction theory for exponential sums over sets of number theoretic interest, such as primes, smooth numbers, and k-th powers, and their applications to analytic number theory and additive combinatorics, including Roth-type theorems in primes and Waring-type results in smooth k-th powers.

Thursday, April 12, 2018

2:00 pm in 241 Altgeld Hall,Thursday, April 12, 2018

#### Spectral Theory on the Modular Surface

Abstract: Many questions about number fields can be recast as questions regarding Laplace eigenvalues on certain manifolds. In this talk I’ll discuss some ideas and results related to Selberg’s trace formula, how partial results towards Selberg’s $\frac{1}{4}$-conjecture have immediate applications, and why number theorists might care about the analysis of Laplacians to begin with.

Thursday, April 26, 2018

2:00 pm in 241 Altgeld Hall,Thursday, April 26, 2018

#### The p-torsion of Ree curves

###### Dane Skabelund (UIUC)

Abstract: This talk will describe some recent computations involving the structure of 3-torsion of the Ree curves, which are a family of supersingular curves in characteristic 3.

Thursday, August 30, 2018

2:00 pm in 241 Altgeld Hall,Thursday, August 30, 2018

#### Organizational Meeting

###### Kyle Pratt (UIUC)

Abstract: We will have a short meeting to draw up a schedule of speakers for the semester. Please sign up if you have a topic in mind! All are welcome, and cookies will be provided.

Thursday, September 13, 2018

2:00 pm in 241 Altgeld,Thursday, September 13, 2018

#### The Combinatorics of Second Order Mock Theta Functions

###### Hannah Burson (University of Illinois at Urbana-Champaign)

Abstract: Less than a year before his death, Ramanujan wrote a letter to Hardy introducing mock theta functions and listing 17 examples of such functions. Zwegers' 2002 thesis allowed for the discovery of infinite families of mock theta functions. The second order mock theta functions are an example of more recently discovered mock theta functions. This talk will introduce new combinatorial interpretations of a general second-order mock theta function identity. In the process, we will talk about some famous results in the theory of partitions and some of the history of Ramanujan.

Thursday, September 27, 2018

2:00 pm in 241 Altgeld Hall,Thursday, September 27, 2018

#### alpha-odd continuation fractions

###### Claire Merriman (University of Illinois at Urbana-Champaign)

Abstract: Nakada’s alpha-expansions move from the regular continued fractions (alpha=1), Hurwitz singular continued fractions (obtained at alpha=little golden ratio), and nearest integer continued fractions (alpha=1/2). This talk will look at similar continued fraction expansions where all of the denominators are odd. I will describe how restricting the parity of the partial quotients changes the Gauss map and natural extension domain. This is join work with Florin Boca.

Thursday, October 4, 2018

2:00 pm in 241 Altgeld Hall,Thursday, October 4, 2018

#### The distribution of elementary symmetric polynomials over finite fields

###### Oscar E. Gonzalez (Illinois Math)

Abstract: The elementary symmetric polynomial of degree k in n variables is formed by adding all distinct products of k distinct variables. In 1950, N. J. Fine proved that these polynomials (mod p) have an asymptotic distribution and gave precise results in the cases p=2 and p= 3. In this talk we will discuss several results and conjectures about the distribution of elementary symmetric polynomials over finite fields, with special attention to the finite field of two elements.

Thursday, October 11, 2018

2:00 pm in 241 Altgeld Hall,Thursday, October 11, 2018

#### Combinatorial methods for ergodic proofs

###### Joseph Vandehey (Ohio State Math)

Abstract: Normal numbers are numbers whose digits display certain typical statistical properties. One early result about normal numbers says that if 0.a_1a_2a_3... is normal, then so is 0.a_ka_{k+\ell}a_{k+2\ell}.... That is, selection along arithmetic progressions preserves normality. By applying deep tools from ergodic theory, Kamae and Weiss have shown that the only sequences along which selection preserves normality are those of low complexity. We will show that part of this result may be proved using combinatorics and analyze these types of problems more broadly.

Thursday, October 18, 2018

2:00 pm in 241 Altgeld Hall,Thursday, October 18, 2018

#### The Fifth Arithmetic Operation

###### Eric Wawerczyk (University of Notre Dame)

Abstract: Martin Eichler is attributed to saying: “There are five elementary operations in Number Theory: addition, subtraction, multiplication, division, and modular forms.” The point of this talk is to demonstrate a variety of amazing arithmetic formulas which can be derived using these five “basic” operations. We will be presenting amazing proofs by Euler, Riemann, and Ramanujan.

Thursday, October 25, 2018

2:00 pm in 241 Altgeld,Thursday, October 25, 2018

#### Ranks of elliptic curves

###### Siegfred Baluyot (Illinois Math)

Abstract: This will be a survey on ranks of elliptic curves over the field of rational numbers. We will review basic definitions about elliptic curves, and then discuss a few open problems and some progress towards them.

Thursday, November 15, 2018

2:00 pm in 241 Altgeld Hall,Thursday, November 15, 2018

#### Partitions with prescribed successive rank parity blocks

###### Ae Ja Yee (Penn State)

Abstract: Successive ranks of a partition, which were introduced by Atkin, are the difference of the i-th row and the i-th column in the Ferrers graph. Recently, in the study of singular overpartitions, George Andrews revisited successive ranks and parity blocks. Motivated by his work, Seunghyun Seo and I investigated partitions with prescribed successive rank parity blocks. In this talk, I will present some results from the collaboration with Seo.

Thursday, November 29, 2018

2:00 pm in 241 Altgeld,Thursday, November 29, 2018

#### Fourier Analysis and the zeros of the Riemann zeta-function

###### Micah B. Milinovich (U. of Mississippi Math)

Abstract: I will show how the classical Beurling-Selberg extremal problem in harmonic analysis arises naturally when studying the vertical distribution of the zeros of the Riemann zeta-function and other L-functions. Using this relationship, along with techniques from Fourier analysis and reproducing kernel Hilbert spaces, we can prove the sharpest known bounds for the number of zeros in an interval on the critical line and we can also study the pair correlation of zeros. Our results on pair correlation extend earlier work of P. X. Gallagher and give some evidence for the well-known conjecture of H. L. Montgomery. This talk is based on a series of papers that are joint with E. Carneiro, V. Chandee, and F. Littmann.