Department of

# Mathematics

Seminar Calendar
for Number Theory Seminar events the year of Tuesday, March 13, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    February 2018            March 2018             April 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                1  2  3    1  2  3  4  5  6  7
4  5  6  7  8  9 10    4  5  6  7  8  9 10    8  9 10 11 12 13 14
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25 26 27 28            25 26 27 28 29 30 31   29 30



Thursday, January 18, 2018

11:00 am in 241 Altgeld Hall,Thursday, January 18, 2018

#### Combinatorial aspects of Levinson's method

###### Nicolas Robles (UIUC Math)

Abstract: The celebrated theorem of Levinson (1974) states that more than 1/3 of the non-trivial zeros of the Riemann zeta-function are on the critical line. This result has been improved during the last 40 years by employing linear and first order terms of a mollifier as well as by using Kloostermania techniques for the error terms. In this work, we delineate how to improve all degrees of the most natural and powerful Dirichlet series (producing an arbitrarily perfect mollification) and we also present the best error terms available with our current technology of exponential sums by elucidating a conjecture of S. Feng. A new and modest % record is thereby achieved. Joint work with Kyle Pratt, Alexandru Zaharescu and Dirk Zeindler.

Thursday, January 25, 2018

11:00 am in 243 Altgeld Hall,Thursday, January 25, 2018

#### The convolution square root of 1 and application to the prime number theorem.

###### Harold Diamond (University of Illinois)

Abstract: We explain what this arithmetic function is and show how the PNT can be deduced from knowledge of its summatory function.

2:00 pm in 241 Altgeld Hall,Thursday, January 25, 2018

#### Decomposition results and the sum product phenomenon

###### George Shakan   [email] (UIUC)

Abstract: Balog and Wooley in 2015 showed that any set can be decomposed into two sets, one that is not additively structured and one that is not multiplicatively structured. I'll talk about some recent progress in the area. As an application, I'll show a bound for the difference-quotient problem as well as mention an application to a character sum.

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

11:00 am in 241 Altgeld Hall,Thursday, February 8, 2018

#### The theta operator and eta-quotients

###### Byungchan Kim (Seoul Technical University )

Abstract: Eta-quotients are a important explicit class of modular forms, and the theta-operator is a natural operator on modular forms. We investigate which eta-quotients are preserved by the theta-operator. This is motivated by a particular partition congruence. This talk is based on joint work with P.-C. Toh and with D. Choi and S. Lim.

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

11:00 am in 241 Altgeld Hall,Thursday, February 15, 2018

#### Potential automorphy and applications

###### Patrick Allen (University of Illinois)

Abstract: The philosophy of Langlands reciprocity predicts that many L-functions studied by number theorists should be equal to L-functions coming from automorphic forms. This leads to Langlands's functoriality conjecture, which very roughly states L-functions naturally created from a given automorphic L-function should also be automorphic. I'll describe this in the case of symmetric power L-functions and how the Langlands program in this special case has applications to the Sato-Tate conjecture and the Ramanujan conjecture. The former concerns the distribution of points on elliptic curves modulo various primes, while the latter concerns the size of the Fourier coefficients of modular forms. I'll then discuss joint work in progress with Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne establishing a weak form of Langlands reciprocity and functoriality for symmetric powers of certain rank 2 L-functions over CM fields.

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

11:00 am in 241 Altgeld Hall,Thursday, February 22, 2018

#### Euler Systems and Special Values of L-functions

###### Corey Stone (University of Illinois)

Abstract: In the 1990s, Kolyvagin and Rubin introduced the Euler system of Gauss sums to derive upper bounds on the sizes of the p-primary parts of the ideal class groups of certain cyclotomic ﬁelds. Since then, this and other Euler systems have been studied in order to analyze other number-theoretic structures. Recent work has shown that Kolyvagin’s Euler system appears naturally in the context of various conjectures by Gross, Rubin, and Stark involving special values of L-functions. We will discuss these Euler systems from this new point of view as well as a related result about the module structure of various ideal class groups over Iwasawa algebras.

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

11:00 am in 241 Altgeld Hall,Thursday, March 1, 2018

#### Monotonicity properties of L-functions

###### Paulina Koutsaki (University of Illinois)

Abstract: In this talk, we discuss some monotonicity results for a class of Dirichlet series. The fact that $\zeta'(s)$ is in that class presents a first example of an arithmetic function for which the associated Dirichlet series is completely monotonic, but not logarithmically completely monotonic. Moreover, we will see how our methods give rise to another formulation of the Riemann Hypothesis for the L-function associated to the Ramanujan-tau function. Based on joint work with S. Chaubey and A. Zaharescu.

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

11:00 am in Siebel 1103,Thursday, March 8, 2018

#### Some generalizations of prime number races problems

###### Xianchang Meng (McGill Math)

Abstract: Chebyshev observed that there seems to be more primes congruent to 3 mod 4 than those congruent to 1 mod 4, which is known as Chebyshev’s bias. In this talk, we introduce two generalizations of this phenomenon. 1) Greg Martin conjectured that the difference of the summatory function of the number of prime factors over integers less than x from different arithmetic progressions will attain a constant sign for sufficiently large x. Under some reasonable conjectures, we give strong evidence to support this conjecture. 2) We introduce the function field version of Chebyshev’s bias. We consider the distribution of products of irreducible polynomials over finite fields. When we compare the number of such polynomials among different arithmetic progressions, new phenomenon will appear due to the existence of real zeros for some associated L-functions.

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

#### Cancelled

Tuesday, March 13, 2018

11:00 am in 241 Altgeld Hall,Tuesday, March 13, 2018

#### Monotonicity properties of L-functions

###### Paulina Koutsaki (UIUC Math)

Abstract: In this talk, we discuss some monotonicity results for a class of Dirichlet series. The fact that $\zeta'(s)$ is in that class presents a first example of an arithmetic function for which the associated Dirichlet series is completely monotonic, but not logarithmically completely monotonic. Moreover, we will see how our methods give rise to another formulation of the Riemann Hypothesis for the L-function associated to the Ramanujan-tau function. Based on joint work with S. Chaubey and A. Zaharescu.

Thursday, March 15, 2018

11:00 am in 241 Altgeld Hall,Thursday, March 15, 2018

#### Some conjectural properties of coefficients of cyclotomic polynomials

###### Bogdan Petrenko (Eastern Illinois U. Math.)

Abstract: The goal of this talk is to interest the audience in some puzzling experimental observations about the asymptotic behavior of coefficients of cyclotomic polynomials. It is well known that any integer is a coefficient of some cyclotomic polynomial. We find it intriguing that when various families of coefficients of cyclotomic polynomials are plotted on the computer screen, the resulting pictures appear "asymptotically almost symmetric". At present, we do not have any theoretical explanation of this perceived behavior of the coefficients. This talk is based on my joint work in progress with Brett Haines (Wolfram Research), Marcin Mazur (Binghamton University), and William Tyler Reynolds (University of Iowa).

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.

Tuesday, March 27, 2018

11:00 am in 241 Altgeld Hall,Tuesday, March 27, 2018

#### Colored Jones polynomials and modular forms

###### Jeremy Lovejoy (Paris 7)

Abstract: In this talk I will discuss joint work with Kazuhiro Hikami, in which we use Bailey pairs and the Rosso-Jones formula to compute the cyclotomic expansion of the colored Jones polynomial of a certain family of torus knots. As an application we find quantum modular forms dual to the generalized Kontsevich-Zagier series. As another application we obtain formulas for the unified WRT invariants of certain 3-manifolds, some of which are mock theta functions. I will also touch on joint work with Robert Osburn, in which we compute a formula for the colored Jones polynomial of double twist knots.

Thursday, March 29, 2018

11:00 am in 241 Altgeld Hall,Thursday, March 29, 2018

#### Around Vinogradov's three primes theorem

###### Fernando Shao (U Kentucky Math)

Abstract: Vinogradov showed in 1937 that every large enough odd integer can be represented as a sum of three primes. One may ask what if these primes are restricted to some (potentially sparse) subset of the primes. In general, if the set is badly distributed in congruence classes or Bohr sets, the result does not necessarily hold. In this talk I will describe two "transference type'' results aimed to show the obstructions described above are the only obstructions. As applications, we get that Vinogradov's three primes theorem holds for Chen primes and for primes in short intervals. This is based on joint works with K. Matomaki and J. Maynard.

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 5, 2018

11:00 am in 241 Altgeld Hall,Thursday, April 5, 2018

#### Large gaps in sieved sets

###### Kevin Ford (UIUC Math)

Abstract: For each prime $p\le x$, remove from the set of integers a set $I_p$ of residue classed modulo $p$, and let $S$ be the set of remaining integers. As long as $I_p$ has average 1, we are able to improve on the trivial bound of $\gg x$, and show that for some positive constant c, there are gaps in the set $S$ of size $x(\log x)^c$ as long as $x$ is large enough. As a corollary, we show that any irreducible polynomial $f$, when evaluated at the integers up to $X$, has a string of $\gg (\log X)(\log\log X)^c$ consecutive composite values, for some positive $c$ (depending only on the degree of $f$). Another corollary is that for any polynomial $f$, there is a number $G$ so that for any $k\ge G$, there are infinitely many values of $n$ for which none of the values $f(n+1),\ldots,f(n+k)$ are coprime to all the others. For $f(n)=n$, this was proved by Erdos in 1935, and currently it is known only for linear, quadratic and cubic polynomials. This is joint work with Sergei Konyagin, James Maynard, Carl Pomerance and Terence Tao.

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 19, 2018

11:00 am in 241 Altgeld Hall,Thursday, April 19, 2018

#### Higher order energy decompositions and the sum-product phenomenon.

###### George Shakan (Illinois Math)

Abstract: In 1983, Erdos and Szemeredi conjectured that either $|A+A|$ or $|AA|$ is at least $|A|^2$, up to a power loss. We make progress towards this conjecture by using various energy decomposition results, in a similar spirit to the recent Balog-Wooley decomposition. Our main tool is the Szemeredi-Trotter theorem from incidence geometry. For more information, see my blog which contains a video introduction the subject: gshakan.wordpress.com

Thursday, April 26, 2018

11:00 am in 241 Altgeld Hall,Thursday, April 26, 2018

#### The Unreasonable Effectiveness of Benford's Law in Mathematics

###### A J Hildebrand and Junxian Li (Illinois Math)

Abstract: We describe work with Zhaodong Cai, Matthew Faust, and Yuan Zhang that originated with some unexpected experimental discoveries made in an Illinois Geometry Lab undergraduate research project back in Fall 2015. Data compiled for this project suggested that Benford's Law (an empirical "law" that predicts the frequencies of leading digits in a numerical data set) is uncannily accurate when applied to many familiar mathematical sequences. For example, among the first billion Fibonacci numbers exactly 301029995 begin with digit 1, while the Benford prediction for this count is 301029995.66. The same holds for the first billion powers of 2, the first billion powers of 3, and the first billion powers of 5. Are these observations mere coincidences or part of some deeper phenomenon? In this talk, which is aimed at a broad audience, we describe our attempts at unraveling this mystery, a multi-year research adventure that turned out to be full of surprises, unexpected twists, and 180 degree turns, and that required unearthing nearly forgotten classical results as well as drawing on some of the deepest recent work in the area.

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.

Tuesday, May 1, 2018

11:00 am in 241 Altgeld Hall,Tuesday, May 1, 2018

#### Average non-vanishing of Dirichlet L-functions at the central point

###### Kyle Pratt (Illinois Math)

Abstract: One expects that an L-function vanishes at the central point either for either deep arithmetic reasons, or for trivial reasons. The central values of Dirichlet L-functions have no arithmetic content, and are also not forced to vanish by the functional equation. One is then led to believe that these central values never vanish, which is a conjecture going back in one form or other to Chowla. The Generalized Riemann Hypothesis implies that almost half of these central values are nonzero. In this talk I will discuss my recent work on central values of Dirichlet L-functions. The main theorem, an unconditional result, is beyond the reach of the Generalized Riemann Hypothesis.