AY 2020-2021

This fall all seminars will be held on Zoom.  There will also be a 30 minute “coffee break” on Zoom after each talk.  Recordings of the talks will be available here.  Several of the talks are planned to tie in with the themes of upcoming mock theta conference being held this Spring.

The Zoom link for the seminar is here. The password is the weight of the Delta function.

Recordings of previous talks can be found on the seminar YT page here: https://www.youtube.com/@vanderbiltnumbertheory6986/videos

Date Speaker, Title, Abstract
Aug 18 Tu Aug 18, 2020 @ 11:00am central time

Dennis Stanton (University of Minnesota) – Historical remarks and recent conjectures for integer partitions.

I will concentrate on two areas:

(1) ranks, cranks, and the Ramanujan congruences for p(n),

(2) the Rogers-Ramanujan identities and MacMahon’s combinatorial versions.

Several open questions will be presented.

The slides can be found here.
Aug 25 Tu Aug 25, 2020 @ 11:00am central time
Martin Raum (Chalmers University of Technology and University of Gothenburg) – Relations among Ramanujan-type Congruences.
We present a new framework to access relations among Ramanujan-type congruences of a weakly holomorphic modular form. The framework is strong enough to apply to all Shimura varieties, and covers half-integral weights if unary theta series are available. We demonstrate effectiveness in the case of elliptic modular forms of integral weight, where we obtain a characterization of Ramanujan-type congruences in terms of Hecke congruences. Finally, we showcase concrete computer calculations, exploring the information encoded by our framework in the case of elliptic modular forms of half-integral weight. This leads to an unexpected dichotomy between Ramanujan-type congruences found by Atkin and by Ono, Ahlgren-Ono.The slides can be found here.
Sep 1 Tu Sep 1, 2020 @ 11:00am central time
Maryam Khaqan (Emory University) – Elliptic Curves and Moonshine.

Moonshine began as a series of numerical coincidences connecting finite groups to modular forms. It has since evolved into a rich theory that sheds light on the underlying structures that these coincidences reflect.
We prove the existence of one such structure, a module for the Thompson group, whose graded traces are specific half-integral weight weakly holomorphic modular forms. We then proceed to use this module to study the ranks of certain families of elliptic curves. This serves as an example of moonshine being used to answer questions in number theory.
This talk is based on arXiv: 2008.01607, where we classify all such Thompson-modules where the graded dimension is a specific weakly-holomorphic modular form and prove more subtle results concerning geometric invariants of certain families of elliptic curves. Time permitting, we will talk about some of these results as well.
The slides can be found here.
Sep 8 Tu Sep 8, 2020 @ 11:00am central time
Nicolas Allen Smoot (RISC) – Partition congruences and the localization method.  A notable problem in partition theory is the study of infinite families of partition congruences modulo powers of a prime.  It has recently been discovered that there exist congruence families, associated with a modular curve of genus 0, for which the traditional methods of proof fail.  One such congruence family is related to the spt analogue of the omega mock theta function.  We recently gave a proof of this congruence family by a new method, based on the manipulation of a localized polynomial ring, rather than by studying $latex \mathbb{Z}[X]$ via the more classical methods.  We will give a brief outline of this method, its surprisingly unique characteristics, and its potential for future work.The slides can be found here.
Sep 15 Tu Sep 15, 2020 @ 11:00am central time
Walter Bridges (Louisiana State University) – Statistics for partitions and unimodal sequences.

The study of the asymptotic distribution of statistics for partitions lies at a crossroads of classical methods and the more recent probabilistic framework of Fristedt and others.  We discuss two results—one that uses the probabilistic machinery and one that calls for a more direct “elementary” method.
We first review Fristedt’s conditioning device and, following Romik, implement a similar construction to give an asymptotic formula for distinct parts partitions of $latex n$ with largest part bounded by $latex t\sqrt{n}$.  We discuss the intuitive advantages of this approach over a classical circle method/saddle-point method proof.
We then turn to unimodal sequences, a generalization of partitions where parts are allowed to increase and then decrease.  We use an elementary approach to prove limit shapes for the diagrams of strongly, semi-strict and unrestricted unimodal sequences.  We also recover a limit shape for overpartitions via a simple transfer.
The slides can be found here.
Sep 22 Tu Sep 22, 2020 @ 11:00am central time
Gene Kopp (University of Bristol) – Indefinite zeta functions.
Indefinite theta functions were introduced by Sander Zwegers in his thesis, in which they are used to generalize and explain the remarkable properties of Ramanujan’s mock theta functions. In this talk, we will discuss the Mellin transforms of indefinite theta functions, which we call indefinite zeta functions. Indefinite zeta functions satisfy a functional equation and live in a continuous parameter space. Special points in this parameter space yield arithmetically interesting zeta functions, such as certain differences of ray class zeta functions of real quadratic fields. Generally, however, indefinite zeta functions are not Dirichlet series but have a series expansion involving hypergeometric functions. We prove a Kronecker limit formula in dimension 2 for indefinite zeta functions as s=0, which specializes to a new analytic formula for Stark class invariants.The slides can be found here.
Sep 29 Tu Sep 29, 2020 @ 11:00am central time
Hannah Burson (University of Minnesota) – Mock theta functions, false theta functions, and weighted odd Ferrers diagrams.
Odd Ferrers diagrams are an analogue of integer partitions that were first introduced by George Andrews as a combinatorial interpretation of $latex \displaystyle q\omega(q)=\sum_{n=0}^\infty \frac{q^{n+1}}{(q;q^2)_{n+1}}$. In this talk, we explore the generating functions that result from attaching various weights to these diagrams.  In the process, we give new combinatorial interpretations of some of Ramanujan’s false theta function identities and two second-order mock theta functions.
Oct 6 Tu Oct 6, 2020 @ 11:00am central time
Nikos Diamantis (The University of Nottingham) – Modular iterated integrals associated with cusp forms.
We show how to produce new types of modular iterated integrals originating in cusp forms and are thus more likely to have arithmetic applications. Further, we introduce a new approach to the characterisation of modular iterated integrals as invariant versions of (linear combinations of) iterated integrals formed by modular forms. These constructions are based on an extension of higher-order modular forms which is of independent interest.The slides are here.
Oct 13 Tu Oct 13, 2020 @ 11:00am central time
Jeremy Rouse (Wake Forest University) – Integers represented by positive-definite quadratic forms and Petersson inner products.
We give a survey of results about the problem of determining which integers are represented by a given quaternary quadratic form Q. A necessary condition for Q(x1,x2,x3,x4) to represent n is for the equation Q(x1,x2,x3,x4) = n to have a solution with x1,x2,x3,x4 in Z_p for all p. But even when n is sufficiently large, this is not sufficient for Q to represent n. The form Q is anisotropic at the prime p if for x1,x2,x3,x4 in Z_p, Q(x1,x2,x3,x4) = 0 implies that x1=x2=x3=x4=0. Suppose that A is the Gram matrix for Q and D(Q) = det(A). We show that if n >> D(Q)^{6+\epsilon}, n is locally represented by Q, but Q fails to represent n, then there is an anisotropic prime p so that p^2 | n and np^{2k} is not represented by Q for any k >= 1. We give sharper results when D(Q) is a fundamental discriminant and discuss applications to universality theorems like the 15 and 290 theorems of Bhargava and Hanke.The slides can be found here.
Oct 27 Tu Oct 27, 2020 @ 11:00am central time
Robert Schneider (University of Georgia) – A multiplicative theory of (additive) partitions.
Much like the natural numbers $latex \mathbb N$, the set $latex \mathcal P$ of integer partitions ripples with interesting patterns and relations. Now, Euler’s product formula for the zeta function as well as his generating function formula for the partition function $latex p(n)$ share a common theme, despite their analytic dissimilarity: expand a product of geometric series, collect terms and exploit arithmetic structures in the terms of the resulting series. Given this correspondence, one wonders: might some theorems of classical multiplicative number theory arise as images in $latex \mathbb N$ of greater algebraic or set-theoretic structures in $latex \mathcal P$, and vice versa?We show that many well-known objects from elementary and analytic number theory can be viewed as special cases of phenomena in partition theory such as: a multiplicative arithmetic of partitions that specializes to much of elementary number theory; a class of “partition zeta functions” containing the Riemann zeta function and other Dirichlet series (as well as exotic non-classical cases); partition-theoretic methods to compute arithmetic densities of subsets of $latex \mathbb N$ as limiting cases of $latex q$-series; and other phenomena at the intersection of the additive and multiplicative branches of number theory.The slides can be found here.
Nov 3 Tu Nov 3, 2020 @ 11:00am central time
Madeline Locus Dawsey (The University of Texas at Tyler) – Modular Forms and Ramsey Theory.
For a fixed integer $latex k \geq 2$, let $latex q$ be a prime power such that $latex q=1 \pmod{k}$ if $latex q$ is even, or $latex q = 1 \pmod{2k}$ if $latex q$ is odd. The generalized Paley graph $latex G_k(q)$ is the graph with vertex set $latex \mathbb{F}_q$ and edge set $latex \{ab : a-b \text{ is a k-th power residue modulo} \  q\}$. We give a formula in terms of finite field hypergeometric functions for the number of complete subgraphs of order four contained in $latex G_k(q)$, and a formula in terms of Jacobi sums for the number of complete subgraphs of order three contained in $latex G_k(q)$, for all k. This generalizes work of Evans, Pulham, and Sheehan on the Paley graph with $latex k=2$. We apply these formulas to obtain lower bounds for the multicolor diagonal Ramsey numbers $latex R_k(4)$ and $latex R_k(3)$. We also examine deep connections to Fourier coefficients of modular forms and rational points on elliptic curves, when q is prime.The slides can be found here.
Nov 9 M Nov 9, 2020 @ 10:00am central time
William Craig (The University of Virginia) – Variants of Lehmer’s Conjecture.
In the spirit of Lehmer’s unresolved speculation on the nonvanishing of Ramanujan’s tau-function, it is natural to ask whether a fixed integer is a value of $latex \tau(n)$ or is a Fourier coefficient $latex a_f(n)$ of any given newform $latex f(z)$. We offer a method, which applies to newforms with integer coefficients and trivial residual mod $latex 2$ Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $latex 3 \leq \ell \leq 37$ are not absolute values of coefficients of newforms with integer coefficients.We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $latex \ell$, we prove that $latex \pm \ell m$ is not a coefficient of any such newform $latex f$ with weight $latex 2k > M_{\pm}(\ell,m) = O_{\ell}(m)$ and even level coprime to $latex \ell$, where $latex M_{\pm}(\ell,m)$ is effectively computable.The slides can be found here.
Nov 17 Tu Nov 17, 2020 @ 11:00am central time
Robert Osburn (University College Dublin) – Generalized Fishburn numbers and torus knots.

The Fishburn numbers are a sequence of positive integers with numerous combinatorial interpretations and interesting asymptotic properties. In 2016, Andrews and Sellers initiated the study of arithmetic properties of these numbers. In this talk, we discuss a generalization of this sequence using knot theory. This is joint work with Colin Bijaoui (McMaster), Hans Boden (McMaster), Beckham Myers (Harvard),  Will Rushworth (McMaster), Aaron Tronsgard (Toronto) and Shaoyang Zhou (Vanderbilt).
The slides can be found here.
Nov 24 Tu Nov 24, 2020 @ 11:00am central time
Ankush Goswami (RISC) – Arithmeticity and quantum modularity for generalized Kontsevich-Zagier strange series.
There has been significant recent interest in the arithmetic
properties of the coefficients of $latex F(1-q)$ and
$latex \mathscr{F}_t(1-q)$ where $latex F(q)$ is the Kontsevich-Zagier strange
series and $latex \mathscr{F}_t(q)$ is the strange series associated to a family
of torus knots. In this talk, we discuss prime power congruences for two
families of generalized Fishburn numbers and the quantum modularity of
$latex \mathscr{F}_t(q)$.
Dec 1 Tu Dec 1, 2020 @ 11:00am central time
Lola Thompson (Utrecht University) – Counting quaternion algebras, with applications to spectral geometry.
We will introduce some classical techniques from analytic number theory and show how they can be used to count quaternion algebras over number fields subject to various constraints. Because of the correspondence between maximal subfields of quaternion algebras and geodesics on arithmetic hyperbolic manifolds, these counts can be used to produce quantitative results in spectral geometry. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack.
Dec 8 Tu Dec 8, 2020 @ 11:00am central time
Amanda Folsom (Amherst College) – Eisenstein series, cotangent-zeta sums, knots, and quantum modular forms.
Quantum modular forms, defined in the rationals Q, transform like modular forms do on the upper half plane H, up to suitably analytic error functions.  In this talk we give frameworks for two different examples of quantum modular forms originally due to Zagier:  the Dedekind sum, and a certain q-hypergeometric sum due to Kontsevich.  For the first, we extend work of Bettin and Conrey and define twisted Eisenstein series, study their period functions, and establish quantum modularity of certain cotangent-zeta sums.  For the second, we discuss results due to Hikami, Lovejoy, the author, and others, on quantum modular and quantum Jacobi forms ultimately related to colored Jones polynomials for a certain family of knots.

DATE SPEAKER, TITLE, ABSTRACT
Feb 3 W Feb 3, 2021 @ 11:00am central time

Jesse Thorner (University of Illinois) – An approximate form of Artin’s holomorphy conjecture and nonvanishing of Artin L-functions.

I will present some recent work with Robert Lemke Oliver
and Asif Zaman in which we noticeably expand the region in which
almost all Artin L-functions in certain families are holomorphic and
nonvanishing.  This combines Galois theory, character theory, and
analytic number theory.  These results are motivated by a wide range
of classically flavored applications; I will focus on applications to
the study of class groups of high-degree number fields.
Feb 10 W Feb 10, 2021 @ 11:00am central time
Sarah Peluse (Princeton/IAS) – Modular zeros in the character table of the symmetric group.
In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $latex p$ the density of entries in the character table of $latex S_n$ that are divisible by $latex p$ goes to $latex 1$ as $latex n$ goes to infinity. I’ll describe a proof of this conjecture, which is joint work with K. Soundararajan. I will also discuss the (still open) problem of determining the asymptotic density of zeros in the character table of $latex S_n$, where it is not even clear from computational data what one should expect.
Feb 17 W Feb 17, 2021 @ 11:00am central time

Joshua Males (University of Cologne) – Cycle integrals, theta lifts, and modular forms.

Cycle integrals are intricately linked to many areas of maths. For example, they encode special values of L-functions, give loop amplitudes in string theory, and appear in algebraic geometry. They can often be realised as certain theta lifts. In this talk I’ll give an overview of recent developments in the use of generalised modular forms in determining rationality results of such cycle integrals. Inspired by breakthrough works of Bringmann-Kane-Kohnen and Bruinier-Ehlen-Yang, I will describe some new results on a certain theta lift, and its relationship to cycle integrals, and how it can be realised in terms of coefficients of generalised modular forms. For example, we see how one can recover relationships to Hurwitz class numbers, or the classical spt partition function.

I will also briefly discuss some related ongoing and future topics. Parts of this talk are based on work with Alfes-Neumann, Bringmann, and Schwagenscheidt as well as Scharf and Schwagenscheidt.
Feb 24 W Feb 24, 2021 @ 10:00am central time

Kathrin Bringmann (University of Cologne) – False theta functions and their modularity properties.

In my talk I will explain how to embed false theta functions into a modular framework and discuss applications.
Mar 17 W Mar 17, 2021 @ 11:00am central time

Jan-Willem Van Ittersum (Utrecht University) – Partitions and quasimodular forms: variations on the Bloch-Okounkov theorem.

Partitions of integers and (quasi)modular forms are related in many ways. We discuss a connection made by a certain normalized generating series of functions f on partitions, called the q-bracket of f. There are many families of functions on partitions, such as (i) the shifted symmetric functions, (ii) their p-adic generalizations, (iii) the weighted t-hook functions and (iv) symmetric functions on partitions, for which the corresponding q-brackets are quasimodular forms. We explain how these four examples can be traced back to the generating series of shifted symmetric functions. The main technical tool for doing so is the study of the Taylor coefficients of strictly meromorphic quasi-Jacobi forms around rational lattice points.
Mar 24 W Mar 24, 2021 @ 11:00am central time
Ling Long (LSU) – A Whipple formula revisited.

A well-known formula of Whipple relates certain hypergeometric values $latex _7F_6(1)$ and $latex _4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $latex HD$, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple’s formula when the hypergeometric data $latex HD$ are primitive and defined over rationals. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms.  We further relate the  hypergeometric values $latex _7F_6(1)$ in Whipple’s formula to the  periods of modular forms.
This is a joint project with Wen-Ching Winnie Li and Fang-Ting Tu. Most of the talk will be on a general background about how different hypergeometric aspects can be fit together.  It should be accessible to graduate students.
Mar 31 W Mar 31, 2021 @ 11:00am central time
Shashika Petta Mestrige (LSU) – Congruences for some partition functions modulo prime powers.

Ramanujan, Watson, Atkin, Gordon, and Hughes used modular functions and modular equations to prove remarkable congruences of the partition function $latex p(n)$ and multi-partitions $latex p_k(n)$. By extending their ideas, we proved the congruences for two parameter family of partitions $latex p_{[1^c\ell^d]}(n)$ modulo powers of $latex \ell$ where $latex \ell$ is a prime $latex (5\leq\ell\leq17)$. We define these partitions by

$latex \displaystyle{\sum_{n=0}^{\infty}p_{[1^c{\ell}^d]}(n)q^n=\prod_{n=1}^{\infty}\frac{1}{(1-q^n)^c(1-q^{\ell n})^d}.}$

Then we used them to derive congruences and incongruences for $latex \ell$-regular partitions, $latex \ell$-core partitions, and $latex \ell$-colored generalized Frobenius partitions.

Then we investigated the congruences modulo arbitrary prime powers by studying a $latex \ell$-adic module associated to the partitions $latex p_{[1^c\ell^d]}(n)$. Our work has been inspired by the work of Folsom-Kent-Ono-Boylan-Webb.
Apr 7 W Apr 7, 2021 @ 11:00am central time

Wei-Lun Tsai (University of Virginia) – Equidistribution of Fourier coefficients of weak Maass forms.

In this talk, I will discuss recent work in which we show that the normalized Fourier coefficients of weak Maass forms of prime level $latex p$ become equidistributed on $latex [-1,1]$ as $latex p \rightarrow \infty$. For integral weight forms, these coefficients are equidistributed with respect to the Sato-Tate measure, while for half-integral weight forms, these coefficients are equidistributed with respect to the arc length measure. The proofs involve a blend of geometric and analytic methods. This is joint work with Riad Masri.
Apr 14 W Apr 14, 2021 @ 11:00am central time
Bernhard Heim (RWTH Aachen University) – Rota’s vision and the Lehmer conjecture.
Lehmer’s conjecture on the non-vanishing of the Ramanujan tau-function is framed in the context of the root distribution of certain families of polynomials. This includes D’Arcais, Nekrasov-Okounkov hook length formula, Chebychev and Laguerre polynomials, and  generalizations of the Bessenrodt-Ono inequality.  We also report on recent progress towards the Chern-Fu-Tang conjecture. This is joint work with Neuhauser.The slides can be found here.
Apr 21 W Apr 21, 2021 @ 11:00am central time

Soon-Yi Kang (Kangwon National University) – Arithmetic properties of the Fourier coefficients of weakly holomorphic modular functions of arbitrary level
The canonical basis of the space of modular functions on the modular group of genus zero form a Hecke system.  From this fact, many important properties of modular functions were derived.
In this talk, we show that the Niebur-Poincare basis of the space of Harmonic Maass functions also forms a Hecke system. As its applications, we show several arithmetic properties of modular functions on the higher genus modular curves such as divisibility of Fourier coefficients of modular functions of arbitrary level and arithmetic of divisor polar harmonic Maass forms.
This is a joint work with Daeyeol Jeon and Chang Heon Kim.
Apr 28 W Apr 28, 2021 @ 11:00am central time

Jeremy Lovejoy (CNRS Université de Paris) – Parity bias in partitions

By parity bias in partitions, we mean the tendency of partitions to have more odd parts than even parts.   In this talk we will discuss exact and asymptotic results for $latex p_{e}(n)$ and $latex p_{o}(n)$, which denote the number of partitions of n with more even parts than odd parts and the number of partitions of n with more odd parts than even parts, respectively.  We also discuss some open problems, one of which concerns a q-series with an “almost regular” sign pattern, reminiscent of some notorious q-series found in Ramanujan’s lost notebook.  This is joint work with Byungchan Kim and Eunmi Kim.

The slides can be found here.
May 5 W May 5, 2021 @ 11:00am central time
Caroline Turnage-Butterbaugh (Carleton College) – Gaps between zeros of the Riemann zeta-function
Let $latex 0 < \gamma_1 \le \gamma_2 \le \cdots $ denote the ordinates of the complex zeros of the Riemann zeta-function function in the upper half-plane. The average distance between $latex \gamma_n$ and $latex \gamma_{n+1}$ is $latex 2\pi / \log \gamma_n$ as $latex n\to \infty$. An important goal is to prove unconditionally that these distances between consecutive zeros can much, much smaller than the average for a positive proportion of zeros. We will discuss the motivation behind this endeavor, progress made assuming the Riemann Hypothesis, and recent work with A. Simonič and T. Trudgian to obtain an unconditional result that holds for a positive proportion of zeros.
May 7 F May 7, 2021 @ 11:00am central time
Min Lee (University of Bristol) – Effective equidistribution of rational points on expanding horospheres

The main purpose of this talk is to provide an effective version of a result due to Einsiedler, Mozes, Shah and Shapira, on the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems due to Ratner. Instead, we pursue an alternative strategy based on spectral theory of automorphic forms, Fourier analysis and Weil’s bound for Kloosterman sums which yields an effective estimate on the rate of convergence in the space of (d+1)-dimensional Euclidean lattices.
This is a joint work with D. El-Baz, B. Huang and J. Marklof.
May 12 W May 12, 2021 @ 11:00am central time

Matthew Just (University of Georgia) – Partition Eisenstein series and semi-modular forms

We identify a class of “semi-modular” forms invariant on special subgroups of $latex GL_{2}(\mathbb{Z})$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein- like series summed over integer partitions, and use it to construct families of semi- modular forms. We ask whether other examples exist, and what properties they all share. This is joint work with Robert Schneider.