BARD 5 Lightning Talks

Lightning above Jackson Square in New Orleans. Photo by reddit user kevlarrr.

Lightning talks are short, informal talks on research or expository topics. The length of talks for BARD 5 will be 7 minutes. A list of speakers, with titles and abstracts, is posted below.

Jena Gregory (University of Texas Rio Grande Valley)

Combinatorial statistics witnessing an infinite family of congruences for a sum of partition functions.

In 2007, Kronholm established infinite families of congruences in arithmetic progression, modulo any prime $\ell$ , for $p(n,m)$ , the function enumerating the partitions of $n$ into parts whose sizes come from the set $[m]$ . In 2022, Eichhorn, Kronholm, and Larsen proved there are combinatorial statistics, known as cranks, that witness Kronholm’s infinite families of congruences.

In this talk, we explore an extension of these results and consider cranks witnessing a sum/difference congruence of the form $p(n,m) \pm p(n',m)\equiv 0 \pmod{\ell}$ , where $n'$ is determined by $n$ .

By an analysis of Ehrhart’s $h^{*}$ -vector, we have established that for certain primes and small values of $m$ , there are cranks witnessing this sum/difference congruence.

Avi Mukhopadhyay (University of Florida)

Some new identities for Andrews’ spt-function

George Andrews defined the smallest parts partition function ${\rm spt}(n)$ and found Ramanujan-type congruences for them. Folsom and Ono found an identity for ${\rm spt}$ involving basic hypergeometric series and eta-quotients to determine its parity. We improve Folsom and Ono’s result for $3/2$ weight mock modular forms and find new simple identities for the ${\rm spt}$ function.

J.C. Saunders (Middle Tennessee State University)

Ranks and cranks of the partition function

In this talk, we introduce some congruence relations of the partition function originally due to Ramanujan and how they can be combinatorially explained in terms of ranks and cranks of partitions. If time permits, we will also discuss some of the provoked research on cranks, including my joint work with Dr. Zafer Selcuk Aygin.

Santiago Radi Severo (Texas A&M University)

Fixed point proportion of groups acting on regular trees

Given a polynomial $f$ and a starting value $a_0$ , what is the proportion (Dirichlet density) of prime numbers that divide at least one non-zero element in the forward orbit of $a_0$ ? Given a polynomial $f$ in a finite field, what is the proportion of points in the finite field that are periodic for $f$ ? And if $f$ is defined in a number field and we take its reduction for every prime ideal, how does the proportion of periodic points in each finite field change as the norm of the prime ideal goes to infinity? All these three questions have a common ingredient called the “Fixed-point proportion.” Given a group acting on an infinite rooted tree, the fixed-point proportion measures the proportion of elements of the group fixing at least one infinite path of the tree. Due to its importance in number theory, many questions about the fixed-point proportion have been posed in the last twenty years. For example, in 2008, Rafe Jones asked whether there exist groups acting transitively on each level of the tree, having positive dimension and having positive fixed-point proportion. The question was answered affirmatively for the binary tree by Nigel Boston in 2010.

In my talk, I will present a family of groups having these three properties in larger trees, extending Nigel Boston’s result. Unlike Boston’s example, in this family we can explicitly compute the fixed-point proportion of the groups, something that is generally almost impossible. The significance of this work lies in the scarcity of large groups having positive and calculable fixed-point proportions. We will also observe that this family is not purely group-theoretical, as the iterated Galois group of the polynomial $x^d+1$ is part of the family.

Chirag Singhal (University of Illinois Chicago)

Lifting of elliptic curves

All across different fields of mathematics, we see some sort of lifting theorems. I will talk about one such lifting theorem regarding elliptic curves by Gun and Murty. Basically, we try to see that given a list of elliptic curves $E_p$ for an infinite list of primes $p$ , when do we get an elliptic curve over $\mathbb{Q}$ such that $E \!\!\mod{p}$ is isogenous to $E_p$ for all $p$ .