Rezza Hadian	Purdue University
Vanessa Medina	St. Mary's University
Shayna Wilson-Spiro	University of Pennsylvania
Zongshu Wu	Princeton University

Wei-Lun Tsai

University of South Carolina

Larry Rolen (July 7)	Vanderbilt University	Title: Conjectures of Andrews on partition-theoretic $q$-series Abstract: In a famous 1986 paper, Andrews made a number of conjectures on the signs and growth rate of $q$-series arising from partition theory. Andrews made these based on computer experiments. The first of these functions, the famous function $\sigma(q):=\sum_{n\geq0}\frac{q^{\frac{n(n+1)}2}}{(-q;q)_n}$, had remarkable growth and vanishing behavior which was finally proven by Andrews-Dyson-Hickerson by tying this series to the arithmetic of the field $\mathbb{Q}(\sqrt6)$. Cohen further uncovered that the numerical phenomenon was due to the $q$-series being what we would now call, thanks to work of Lewis-Zagier, a period integral of a Maass waveform. This was also an early example of the new theory of Zwegers mock Maass theta functions, and of a quantum modular form. In the same paper, Andrews also made conjectures on remarkable sign behavior of partition theoretic functions, such as $v_1(q):=\sum_{n\geq0}\frac{q^{\frac{n(n+1)}2}}{(-q^2;q^2)_n}$. I will discuss recent work, joint with Folsom, Males, and Storzer, establishing some of these. I will also discuss recent work by Kalita, Kundu, Storzer, and Wang on related conjectures of Andrews and new conjectural infinite families.
William Craig (July 14)	United States Naval Academy	Title: Integer partitions detect the primes Abstract: This talk presents a partition-theoretic approach to the idea of Diophantine equations motivated by the famous resolution of Hilbert’s Tenth Problem by Matiyasevich. We resolve questions of Schneider on detecting important sets of integers using ’Diophantine equations with partition function’. The Diophantine equations we consider involve equations of partition functions considered by MacMahon and their natural generalizations in the emerging theory of q-multiple zeta values. Here we explicitly construct infinitely many Diophantine equations in partition functions whose solutions are the prime numbers. We conclude that prime numbers can be detected from partitions and symmetric polynomials alone. This is joint work with Jan-Willem van Ittersum and Ken Ono.
Walter Bridges (July 21)	University of North Texas	Title: Integer partitions and random representations of Lie algebras Abstract: Choose a partition of a large integer uniformly at random. How big do we expect the largest part to be? How many 1’s are there? What does the Young diagram look like? For that matter, how can we generate large partitions efficiently, to collect data and make conjectures? In 1993, Fristedt introduced a statistical mechanics inspired approach to these sorts of questions that has proved widely useful in analytic combinatorics. Adapting Fristedt’s conditioning device to our setting, I will describe how a typical, large-dimensional representation looks for the family of complex Lie algebras, $\mathfrak{sl}_2(\mathbb{C})$. (The case r = 1 corresponds to integer partitions.) In particular, we give asymptotic probability distributions for the multiplicity of small irreducible representations, as well as the largest dimension, the largest height, and the total number of irreducible representations appearing in the decomposition of a representation sampled uniformly from all representations with the same dimension. This is joint work with Kathrin Bringmann and Caner Nazaroglu.

Tapas Bhowmik

University of South Carolina

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