Date: October 5, 2022

Speaker: **Hasan Saad** (University of Virginia)

**Matthias Storzer**(MPIM Bonn)

Title: Nahm sums and their modularity

**William Keith**(Michigan Tech)

Abstract: Ramanujan-Kolberg identities relate subprogressions of the

partition numbers and linear combinations of eta-quotients, so named

after Ramanujan’s “most beautiful identity” $\sum_{m=0}^\infinity

p(5m+4) q^n = 5 \prod_{n=1}^\infty \frac{(1-q^{5n})^5}{(1-q^n)^6}$.

Descending from equality to congruence mod 2, recent work of the

speaker and Fabrizio Zanello has produced a large number of these with

implications for the study of the parity of the partition function,

and Shi-Chao Chen has shown that these are part of an infinite family

where the eta-quotients required have a very small basis. Analyses of

particular cases from that family yield many pleasing patterns: older

work of the speaker and Zanello gave congruences for the $m$-regular

partitions for $m$ odd, and this talk will be on more recent joint

work which exhibits the very different behavior for $m$ even which

gives connections to multipartitions. All of these results in turn

illuminate different aspects of the longstanding partition parity

problem and hopefully provide some useful insight therein.

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

Title: A Nonvanishing Result of Kohnen

Abstract: In this talk we will review Kohnen’s short proof of a nonvanishing result for the L-function of a cuspidal Hecke eigenform on the full modular group. We will highlight generalizations in the literature and possible open directions.

Date: February 22, 2023

Speaker: **Daniel Lautzenheiser** (Cerro Coso Community College)

Title: Sphere packing, Fourier eigenfunctions, and modular forms

Abstract:

In this talk, we will discuss how the sphere packing problem can give

rise to specific transformation properties of functions. In some

dimensions, these transformation properties are satisfied by modular

forms. In particular, we look at the case d=4 and exhibit a modular

form toward creating a Fourier eigenfunction with extremal properties.

Date: March 20:

Speaker: **Frank Garvan** (University of Florida)

Title: Cultivating Maple and Sage in Ramanujan’s Garden.

Abstract: This talk is on experimental math and how we can use Maple

and Sage to gain insight into various rank and crank type functions.

Date: March 22

Speaker: **Robert Schneider** (Michigan Tech)

Title: Multiplicative theory of integer partitions

Abstract: I give an incomplete survey of a meta-theory of partitions I began developing in graduate school with my pair of papers on partition zeta functions (2016) and the $q$-bracket (2017), that is still under construction in collaboration with my coauthors and students.

Much like the natural numbers $\mathbb N$, the set $\mathcal P$ 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 $p(n)$ 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. Works of George E. Andrews from the 1970s hint at algebraic and analytic super-structures unifying aspects of partition theory and arithmetic. One wonders then: might some theorems of classical multiplicative number theory arise as images in $\mathbb N$ of greater algebraic or set-theoretic structures in $\mathcal P$?

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 many theorems 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 and Abelian-type theorems as limiting cases of $q$-series; and other phenomena at the intersection of the additive and multiplicative branches of number theory. Includes joint work with Ken Ono, Larry Rolen, Andrew Sills, Ian Wagner and other co-authors.