# Vanderbilt University Number Theory Seminar

This spring 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.

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

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 $$p$$ the density of entries in the character table of $$S_n$$ that are divisible by $$p$$ goes to $$1$$ as $$n$$ 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 $$S_n$$, 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 24 W Mar 24, 2021 @ 11:00am central time
Ling Long (LSU) – TBA
TBA
Apr 7 W Apr 7, 2021 @ 11:00am central time

##### Wei-Lun Tsai (University of Virginia) – TBA

TBA.

Apr 14 W Apr 14, 2021 @ 11:00am central time
Bernhard Heim (RWTH Aachen University) – TBA
TBA
Apr 21 W Apr 21, 2021 @ 11:00am central time

##### Soon-Yi Kang (Kangwon National University) – TBA

TBA.

May 5 W May 5, 2021 @ 11:00am central time
Caroline Turnage-Butterbaugh (Carleton College) – TBA
TBA