Date: October 3, 2023

Speaker: **Andreas Mono **(Vanderbilt University)

Title: A modular framework of functions of Knopp

Abstract: This talk presents the construction of a modular completion of a function introduced by Knopp 30 years ago in his paper Modular integrals and their Mellin transforms. His function is closely related to a term by term lift of Zagier’s influential f_{k,D} function under the Bol operator. We begin with a motivation of the topic, and summarize some background briefly. Afterwards,

we outline our constructions, and discuss their naturality. We connect our result to the more recently introduced concept of locally harmonic Maaß forms by Bringmann, Kane, and Kohnen about 10 years ago. In addition to that, we present connections of our results to some earlier work on hyperbolic Eisenstein series and their local modular completions. The first part is joint work with Kathrin Bringmann.

Slides can be found: here

Date: October 17, 2023 (via Zoom)

Speaker: **Shane Chern** (Dalhousie University)

Title:Asymptotics for nonmodular infinite products and the Seo-Yee conjecture

Abstract: In this talk, I will present my recent work on the asymptotics for a generic family of nonmodular infinite products near an arbitrary root of unity. More precisely, our attention is focused on infinite products of the form $1/(q^a;q^M)_\infty$ where $M$ is a positive integer and $a$ is any of $1, 2, \ldots, M$. Such asymptotic expansions will be utilized to prove a conjecture of Seunghyun Seo and Ae Ja Yee, up to a finite check. The Seo-Yee Conjecture, asserting that the series expansion of a certain infinite product has nonnegative coefficients, is equivalent to the Coll-Mayers-Mayers Conjecture on the index statistic for seaweed algebras.

Date: October 24, 2023

Speaker: **Eleanor McSpirit **(University of Virginia)

Title: Jellyfish, the arithmetic-geometric mean, and elliptic curves

Abstract: In this talk, we discuss a finite-field analogue of the arithmetic-geometric mean sequence AGM over the reals. Study of the classical version dates back to work of Lagrange and Gauss, and makes beautiful contact with approximations of pi, elliptic integrals, hypergeometric functions, and elliptic curves. In the study of \text AGM}(F_q), directed graphs called “jellyfish swarms” naturally arise.

In studying these graphs, we find connections to both finite-field hypergeometric functions and elliptic curves over finite fields. Such connections give rise to new identities for Gauss’ class numbers of positive definite binary quadratic forms and allow us to show that the sizes of “jellyfish” (connected components of these graphs) are in part dictated by the order of the prime above 2 in certain class groups.

Date: November 7, 2024

Speaker: **Scott Ahlgren **(UIUC)

Title: Congruences for the partition function

**Aidan Lorenz**(Vanderbilt University)

Title: Small dilatation pseudo-Anosovs coming from fibered 3-manifolds.

Abstract: Every pseudo-Anosov homeomorphism has an associated number called the dilatation. We will start by discussing some interesting number-theoretic properties and questions about these dilatations. We will then discuss the relationship between pseudo-Anosovs and fibered 3-manifolds, the Thurston norm, and how number theory plays a role in some recent results on the topic.

Date: December 4, 2023:

Speaker: **Karen Taylor** (Bronx Community College, CUNY)

Title: An example of a new class of period functions.

Abstract:In this talk we use Knopp’s construction of a rational period function to give an example of a more general period function.

Date: March 26, 2024

Speaker: **Wei-Lun Tsai** (University of South Carolina)

Title: Even values of Ramanujan’s tau-function

Abstract: Inspired by Lehmer’s conjecture on the non-vanishing of

Ramanujan’s tau-function, it is natural to ask whether any given

integer is a tau-value. Many recent works have identified explicit

examples of odd integers which are not tau-values. In this talk, I

will discuss the examples of even integers that are not tau-values and

demonstrate such results for infinitely many even integers. This is

joint work with Jennifer Balakrishnan and Ken Ono.

Date: April 9, 2024

Speaker: **Kalani Thalagoda** (Tulane University)

Title: Bianchi modular forms over $\mathbb{Q}(\sqrt{-17}$

Abstract:

Bianchi Modular Forms are generalization of classical modular forms defined over Imaginary quadratic fields. Similar to the classical case, we can use the theory of modular symbols for computation. However, when the class group of the Imaginary quadratic field is non-trivial, we can only directly compute certain components. But we can still use some computational tricks to extract the Bianchi Modular forms as Hecke eigensystems. In this talk, I will go over some of these computational techniques for the field $\mathbb{Q}(\sqrt{-17}$ which has class number $4$. With explicit example, I will demonstrate interesting Hecke eigensystems we observed.