Department of

# Mathematics

Seminar Calendar
for Number Theory Seminar events the year of Thursday, October 11, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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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.

Thursday, August 30, 2018

11:00 am in 241 Altgeld Hall,Thursday, August 30, 2018

#### Equal sums of two cubes

###### Bruce Reznick (Illinois Math)

Abstract: Ramanujan was fascinated by equal sums of two cubes, as shown in the legendary anecdote about $1729 = 10^3 + 9^3 = 12^3 + 1^3$. He was also interested in the equation $f_1^3 + f_2^3 = f_3^3 + f_4^3$ for polynomials $f_j$ and gave several examples in which the $f_j$'s were quadratic polynomials. We will discuss the complete solution for quadratic complex polynomials, with side trips into elliptic curves and computational algebraic geometry. The simplest such identity is $(x^2 + x y - y^2)^3 + (x^2 - x y - y^2)^3 = 2x^6 -2y^6$. This talk is meant to be accessible to first-year graduate students.

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 6, 2018

11:00 am in 241 Altgeld Hall,Thursday, September 6, 2018

#### $\alpha$-expansions with odd partial quotients

###### Florin Boca (UIUC)

Abstract: Nakada's $\alpha$-expansions interpolate between three classical continued fractions: regular (obtained at $\alpha=1$), Hurwitz singular (obtained at $\alpha$=little golden mean), and nearest integer (obtained at $\alpha$=1/2). This talk will consider $\alpha$-expansions in the situation where all partial quotients are asked to be odd positive integers. We will describe the natural extension of the underlying Gauss map and the ergodic properties of these transformations. This is joint work with Claire Merriman.

2:00 pm in 241 Altgeld,Thursday, September 6, 2018

#### A survey on primes in arithmetic progressions

###### Kyle Pratt (UIUC)

Abstract: I will give a survey talk on primes in arithmetic progressions. The talk should be accessible to any graduate student, number theorist or not.

Thursday, September 13, 2018

11:00 am in 241 Altgeld Hall,Thursday, September 13, 2018

#### Breaking the 1/2-barrier for the twisted second moment of Dirichlet L-functions

###### Kyle Pratt (University of Illinois)

Abstract: Our seminar lecture on Thursday will be given by Kyle Pratt. His title and abstract are given below. Title: Breaking the 1/2-barrier for the twisted second moment of Dirichlet L-functions Abstract: I will discuss very recent work, joint with Bui, Robles, and Zaharescu, on a moment problem for Dirichlet L-functions. The talk will be accessible, as I will give lots of background information and will not dwell on technicalities.

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 20, 2018

11:00 am in 241 Altgeld Hall,Thursday, September 20, 2018

#### Maass forms and the mock theta function f(q)

###### Scott Ahlgren (Illinois Math)

Abstract: Let f(q) be the well-known third order mock theta function of Ramanujan. In 1964, George Andrews proved an asymptotic formula for the Fourier coefficients of f(q), and he made two conjectures about his asymptotic series (these coefficients have an important combinatorial interpretation). The first of these conjectures was proved in 2009 by Bringmann and Ono. Here we prove the second conjecture, and we obtain a power savings bound in Andrews’ original asymptotic formula. The proofs rely on uniform bounds for sums of Kloosterman sums which follow from the spectral theory of Maass forms of half integral weight and in particular from a new estimate which we derive for the Fourier coefficients of such forms. This is joint work with Alexander Dunn.

Thursday, September 27, 2018

11:00 am in 241 Altgeld Hall,Thursday, September 27, 2018

#### Full-image p-adic Galois Representations and Galois Deformation Theory

###### Shiang Tang (Illinois Math)

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

11:00 am in 241 Altgeld Hall,Thursday, October 4, 2018

#### Dirichlet L-functions of quadratic characters of prime conductor at the central point

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

Abstract: In this joint work with Kyle Pratt, we prove that more than nine percent of the central values $L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1$ (mod $8$) ranges over primes and $\chi_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain estimates for the second and third moments of $L(\frac{1}{2},\chi_p)$.

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

11:00 am in 241 Altgeld Hall,Thursday, October 11, 2018

#### Higher-dimensional frontiers in continued fractions

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

Abstract: Abstract: We will discuss results from a long-running project between myself and Anton Lukyanenko to understand the connections between continued fractions and hyperbolic geometry in higher dimensions. Our recent breakthrough has allowed us to consider the connections between complex continued fractions and three-dimensional real hyperbolic space, quaternionic continued fractions and five-dimensional real hyperbolic space, octonionic continued fractions and nine-dimensional real hyperbolic space, and also Heisenberg continued fractions and two dimensional complex hyperbolic space. We will discuss the implications of these connections on a variety of number theoretic problems, including rational approximation and the study of algebraic irrationals, and many new problems these results have led us to. This talk will have connections to dynamical systems and hyperbolic geometry.

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

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

#### Arithmetic properties of Hurwitz numbers

###### David Hansen (University of Notre Dame)

Abstract: Hurwitz numbers are the "$\mathbf{Q}(i)$-analogue" of Bernoulli numbers; they show a remarkable number of patterns and properties, and deserve to be better-known than they are. I'll discuss some old results on these numbers due to Hurwitz and Katz, and some newer results obtained by four Columbia undergraduates during a summer REU I supervised. No background knowledge will be assumed.

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 1, 2018

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

#### Quantum chaos and arithmetic

###### Simon Marshall (University of Wisconsin-Madison)

Abstract: If M is a compact manifold of negative curvature, Laplace eigenfunctions on M with large eigenvalue are expected to behave chaotically, reflecting the correspondence principle between classical and quantum mechanics. I will describe this chaotic behavior, and explain what can be proved about it using methods from harmonic analysis. I will also explain why harmonic analysis alone has a hard time giving us the complete picture, and how we can see more of it using tools from number theory.

Thursday, November 15, 2018

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

#### Some truncated partition theorems

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

Abstract: The recent work of George Andrews and Mircea Merca on the truncated version of Euler's pentagonal number theorem has opened up a new study on truncated theta series. Since then several papers on the topic have followed. For instance, Victor Guo and Jiang Zeng investigated two theta series of Gauss and proved analogous results to the truncated theorem of Andrews and Merca. In this talk, I will discuss their results along with some new discoveries from my collaboration with Chun Wang.

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

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

#### Fourier optimization and primes in short intervals

###### Micah Milinovich (University of Mississippi Math)

Abstract: I will discuss the proofs of the strongest known asymptotic and explicit estimates in the classical problem of bounding the maximum gap between consecutive primes assuming the Riemann hypothesis. These proofs involve three main ingredients: the explicit formula relating the primes to the zeros of the Riemann zeta-function, the size of the constant in the Brun-Titchmarsh inequality, and Fourier optimization. I will also discuss how to use Fourier optimization, in particular the solution to the Beurling-Selberg extremal problem for the Poisson kernel, to estimate the variance of primes on short intervals sharpening the previous results of Selberg, Montgomery, Gallagher and Mueller, Goldston and Gonek, and others. This talk is based on joint works with E. Carneiro, V. Chandee, A. Chirre, and K. Soundararajan.

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.

Thursday, December 6, 2018

11:00 am in 241 Altgeld Hall,Thursday, December 6, 2018

#### Divisibility of the central binomial coefficients

###### Kevin Ford (Illinois Math)

Abstract: Abstract:That $(n+1)| binom{2n}{n}$ for every positive integer $n$ is a consequence of the integrality of the Catalan numbers. Pomerance conjectured that $n| binom{2n}{n}$ for a set of positive density. In joint work with Sergei Konyagin, we prove Pomerance's conjecture and give an explicit form of the density. The proof uses a new method for accounting for the largest prime factors of integers.