BARD 2 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 2 will be 10 minutes. A list of speakers follows, with titles and abstracts.

John Lopez (Tulane University)

The lexicographically least square-free word with a given prefix.

The lexicographically least square-free infinite word on the alphabet of non-negative integers with a given prefix $p$ is denoted $L(p)$ . When $p$ is the empty word, this word was shown by Guay-Paquet and Shallit to be the so-called ruler sequence. For other prefixes, the structure is significantly more complicated. We show that $L(p)$ reflects the structure of the ruler sequence for several words $p$ . We provide morphisms that generate $L(n)$ for letters $n = 1$ and $n \geq 3$ .

Brian Grove (Louisiana State University)

Hypergeometric Functions, Recursive Sequences, and Special Zeta Values.

Hypergeometric Functions are inherently recursive objects. Therefore, it seems reasonable to expect a connection between special values of terminating hypergeometric functions and well-known recursive sequences, such as the Fibonacci numbers, Catalan numbers, and Bernoulli numbers. I will give examples of such connections. Combining a few of these examples gives a method to compute some well-known special values of the Riemann Zeta function in terms of special values for terminating hypergeometric functions and appropriate multiples of $\pi$ . I will discuss how to compute these special zeta values through examples.

Kalani Thalagoda (University of North Carolina Greensboro)

Computing Modular Forms

Classical modular forms are extremely nice functions on the hyperbolic plane. In particular, they have Fourier expansions and the coefficients of these expansions are theoretically interesting. In this lightning talk, I will discuss some techniques available to compute these Fourier coefficients. The main reference for this talk is William Stein’s Modular Forms: A Computational Approach.

Rahul Kumar (Pennsylvania State University)

Ramanujan’s formula for odd zeta values over number fields

In this talk, we will discuss transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan’s famous identity for the Riemann zeta function. Our transformation formulas have far-reaching implications such as they lead to a new number field extension of Eisenstein series over ${\rm SL}_2(\mathbb{Z})$ . This is joint work with Soumyarup Banerjee and Rajat Gupta.

Emma Lien (Louisiana State University)

Inverses of Polynomials

Given a polynomial $f \in \mathbb{Z}[x]$ , we can view it as a real analytic function $f:\mathbb{R} \to \mathbb{R}$ and extend even further to a holomorphic function $f:\mathbb{C} \to \mathbb{C}$ . The goal is, can we explicitly construct an inverse function $g$ with similar analytic properties, and describe the Galois action on $g(t)$ for any $t \in \mathbb{Q}$ . This talk will focus on the construction of a specific example where such an inverse function exists and discuss the obstructions in generalizing this construction.

Rajat Gupta (University of Texas at Tyler)

Divisor function in number fields

In this talk I will introduce the divisor function associated with a general number field. I will discuss its analytic properties and conclude the talk by showing a closed form evaluation of the partial summation of this divisor function. This is a joint work with Sudip Pandit. He is a final year PhD students at the Indian Institute of Technology Gandhinagar in India.