BARD 7 Lightning Talks

Digital art of a nutria bard summoning lightning from the sky with a piece of chalk. Created by Viv Heurtevant.

Lightning talks are short, informal talks on research or expository topics. The length of talks for BARD 7 will be 8 minutes. A list of speakers, with titles and abstracts, is available below.

O. David Agbolade (Louisiana State University)

Identifying unique constructions of complex equiangular tight frames in small dimensions

Equiangular tight frames (ETFs) represent the optimal solution to a fundamental geometric problem: arranging $n$ lines through the origin in $\mathbb{C}^d$ so that they are as evenly spread as possible. Such frames are known to exist for select pairs $(d, n)$ , yet the question of how many distinct ETF constructions exist for a given pair—up to a suitable notion of equivalence—remains open. We employ the number of distinct triple products, their field of existence, and other invariants to classify ETFs for small parameters $(d \leq 7)$ .

Tung Hoang (Tulane University)

Quadratic Twists with trivial $\ell$ -Selmer groups over totally real fields

We study quadratic twists of elliptic curves over totally real number fields. We show that, under suitable hypotheses, there exist large families of twists with trivial $\ell$ -Selmer group, and hence rank 0. The argument is based on a systematic way of imposing compatible local conditions across primes and assembling them into global families of twists. This is ongoing joint work with my advisor, Professor Olivia Beckwith.

Jayashree Kalita (Vanderbilt University)

Oscillating asymptotics and almost alternating sign patterns

Computer experiments led Andrews, in 1986, to conjecture striking sign patterns and growth phenomena for the coefficients of five partition-theoretic q-series from Ramanujan’s Lost Notebook. The first of these functions, the now famous series

$\displaystyle \sigma(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q;q)_n}$

exhibits remarkable growth and vanishing behavior, which was proven by Andrews, Dyson, and Hickerson by tying this series to the arithmetic of the quadratic field $\mathbb{Q}(\sqrt{6})$ . Cohen further uncovered that the numerical phenomenon was due to the q-series being what we would now call, thanks to the work of Lewis-Zagier, a period integral of a Maass waveform. This example also foreshadowed the modern theories of mock Maass theta functions initiated by Zwegers and quantum modular forms introduced by Zagier.

However, the other four q-series remained largely unexplored until recent work of Folsom, Males, Rolen, and Storzer, who proved some of the Andrews’ conjectures for the series

$\displaystyle v_1(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}.$

In joint work with Debanjana Kundu, Matthias Storzer, and Xintong Wang, we established almost alternating sign patterns for coefficients of the remaining three q-series along with proving a conjecture of Andrews from his 1986 paper. Using analytic techniques such as the method of steepest descent and the circle method, we derived asymptotics for the coefficients, whose alternating and oscillatory behavior explains the observed patterns. We also introduced a new family of q-series exhibiting similar phenomena.

In this talk, I will give a non-technical overview of the main ideas.

Esme Rosen (Louisiana State University)

TBA

Chirag Singhal (University of Illinois Chicago)

TBA

Clayton Williams (University of Illinois Urbana-Champaign)

Powers of the eta function through the Weil representation

The metaplectic group is a double cover of ${\rm SL}_2(\mathbb{Z})$ which allows one to handle the multi-valuedness of the square root for half-integer weight modular forms in a particularly elegant way. As a nonlinear group, it is convenient to consider the Weil representation of the metaplectic group instead of working with the original group. An elementary example of a modular form which can be written as a vector-valued modular form transforming with respect to the Weil representation is the Dedekind eta function. In this case, the lattice attached to the Weil representation is of the simplest form, a rank 1 lattice. While not formally recorded as a conjecture, most mathematicians who worked with the eta function would likely have stated that odd powers of the eta function should also be associated with rank 1 lattices. It is not difficult to show, however, that higher powers of eta actually transform with respect to lattices of higher rank. This leads one to consider Weil representations attached to finite quadratic modules, which can be easier to construct and, in some cases, to work with. In this talk vector-valued modular forms for powers of eta will be presented, and (time permitting) some applications of the finite quadratic module perspective will be discussed.