BARD 3 Lightning Talks

Lightning above Tiger Stadium on the LSU campus. Photo by ESPN.

Lightning talks are short, informal talks on research or expository topics. The length of talks for BARD 3 will be 7 minutes. A list of speakers follows, with titles and abstracts.

Simon Rutard (Institut Camille Jordan, France)

Special values at nonpositive integers of some Witten zêta functions

In this talk, we will define Witten zêta functions associated with semi-simple Lie algebras, and then give a brief understanding about how to compute some of the values, and their derivative values, at nonpositive integers of the Witten zêta functions associated with $\mathfrak{so}(5)$ and $\mathfrak{g}_2$ . By combining our results with a Meinardus-type theorem, we will also give an explicit asymptotic formula for the number of representations of dimension $n$ of the Lie algebra $\mathfrak{g}_2$ , as $n \to +\infty$ .

Brian Grove (Louisiana State University)

The residue sum method for supercongruences

Supercongruences for truncated hypergeometric functions connect the complex analytic, p-adic, and algebraic worlds in interesting ways. In this talk, I will discuss a useful method of proving supercongruences that connects the complex analytic and p-adic worlds using residue sums.

Sinuhe Perea (King’s College London / Max Planck Institute)

TBA

Navtej Singh (University of Michigan)

On the Shioda conjecture for diagonal projective varieties over finite fields

We study algebraic varieties (surfaces) defined by a single polynomial with coefficients in a finite field. Our goal is to understand the one-dimensional algebraic varieties (curves) that lie on such a surface. In particular, we aim to construct surfaces which contain “as many linearly independent curves as possible”; examples of such surfaces are sparse, and it would be quite interesting to have more. Concretely, with regards to the Shioda conjecture, families of supersingular varieties seem to be hard to come by, and the ones often constructed in the literature are already known to be unirational, so we seek families of supersingular varieties whose unirationality is currently unknown as test cases. By some deep theorems and conjectures in algebraic geometry, this problem can be translated into a more concrete one involving numbers of solutions to the defining polynomial over finite fields. This more concrete problem is the one we will try to tackle.

Paresh Singh Arora (Louisiana State University)

The Klein quartic curve and its modularity

The Klein quartic curve is a genus 3 compact Riemann surface with an automorphism group of order 168, which is the highest for its genus. It can be viewed as a projective algebraic curve over complex numbers, defined by the equation $x^3 y + y^3 z + z^3 x = 0$ . In this talk, we calculate the local zeta function of the Klein quartic curve by viewing it as a quotient of degree 7 Fermat curve and find the associated modular forms.

Esme Rosen (Louisiana State University)

Hypergeometric identities arising from Atkin-Lehner involutions

TBA

Archisman Bhattacharjee (Louisiana State University)

Ramanujan’s universal quadratic forms

In this talk, we discuss 54 diagonal quaternary quadratic forms given by Ramanujan. Given a natural number n, we talk about a general method to find the number of representations of n by a specific form. In the end, we explicitly calculate this number for a couple of forms, namely $x^2+y^2+3z^2+3t^2$ and $x^2+y^2+z^2+5t^2$ .

Koustav Mondal (Louisiana State University)

Congruent theta series

The congruent theta series serves as a potent instrument for examining quadratic forms with congruence conditions. This presentation is intended to provide an introduction and illustrate the characteristics of the congruent theta series and examples.

Evangelos Nastas (University of Houston)

Revisiting the distribution of prime numbers

This lightning talk delves into the mesmerizing world of prime numbers and their distribution, a topic that has enamored mathematicians for centuries. The function $\pi(x)$ , defined as the cardinality of prime numbers less than $x$ , is revisited, and its various estimates, including the well-known approximation $x/\ln(x)$ , explored. Then the attention is turned to the logarithmic integral of Gauss, ${\rm li}(x)=\int_2^x \frac {1}{\ln(t)} dt$ , an approximation of $\pi(x)$ whose implications extend to the Riemann Hypothesis (RH) and related problems, admitting multiple forms and having involved a plethora of results in number theory, analysis, operator algebras, etc. The question amounts to locating the zeros of the Riemann zeta function $\zeta(s)$ on ${\rm Re}(s)=1/2$ . It is known that the zeros are symmetrical with respect to ${\rm Re}(s)=1/2$ line to the $\zeta(s)$ . Some of the most optimal approximations of $\pi(x)$ are $O(x \exp(-c \ln(x)^{1/2}))$ . It can be shown that a sufficient approximation of $O(x^{1/2+\epsilon})$ for $\pi(x)$ implies RH. The RH, which is concerned with the zeros of the Riemann zeta function $\zeta(s)$ , needless to mention, as widely known, has profound implications across a plethora of topics in mathematics. The discussion will focus on three main points. First, on computing the reciprocal function of the integral to approximate prime numbers. Second, on demonstrating that a close approximation of $\pi(x)$ implies RH. And finally, on showing that an analytic extension implies RH.

Peter Marcus (Tulane University)

Miller bases for spaces of cusp forms

A Miller basis for a d-dimensional space of cusp forms has the property that the i-th Fourier coefficient of the i-th basis element is 1, all others up to the d-th coefficient are 0, and optionally, all coefficients are integers. Many, but not all, spaces of cusp forms have a Miller basis. I will discuss this problem of existence of Miller bases and their relation to Poincare series and Sturm bounds.