Vanderbilt University Number Theory Seminar

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.

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 \(\mathbb{Z}[X]\) 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 \(n\) with largest part bounded by \(t\sqrt{n}\).  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.
Sep 29 Tu Sep 29, 2020 @ 11:00am central time
Hannah Burson (University of Minnesota) – TBA.
Oct 6 Tu Oct 6, 2020 @ 11:00am central time
Nikos Diamantis (The University of Nottingham) – TBA.
Oct 13 Tu Oct 13, 2020 @ 11:00am central time
Jeremy Rouse (Wake Forest University) – TBA.
Oct 27 Tu Oct 27, 2020 @ 11:00am central time
Robert Schneider (University of Georgia) – TBA.
Nov 3 Tu Nov 3, 2020 @ 11:00am central time
Madeline Locus Dawsey (The University of Texas at Tyler) – TBA.
Nov 9 M Nov 9, 2020 @ 10:00am central time
William Craig (The University of Virginia) – TBA.
Nov 17 Tu Nov 17, 2020 @ 11:00am central time
Robert Osburn (University College Dublin) – TBA.
Nov 24 Tu Nov 24, 2020 @ 11:00am central time
Ankush Goswami (RISC) – TBA.
Dec 1 Tu Dec 1, 2020 @ 11:00am central time
Lola Thompson (Utrecht University) – TBA.
Dec 8 Tu Dec 8, 2020 @ 11:00am central time
Amanda Folsom (Amherst College) – TBA.