BARD 4 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 4 will be between 5 and 10 minutes. A list of speakers, with titles and abstracts, will be posted below.

Navvye Anand (California Institute of Technology)

On bounds and Diophantine properties of elliptic curves

Mordell equations are celebrated equations within number theory and are named after Louis Mordell, an American-born British mathematician, known for his pioneering research in number theory. We discover all Mordell equations of the form $y^2 = x^3 + k$ , where $k \in \mathbb Z$ , with exactly $|k|$ integral solutions. We also discover explicit bounds for Mordell equations, parameterized families of elliptic curves and twists on elliptic curves. Using the connection between Mordell curves and binary cubic forms, we improve the lower bound for number of integral solutions of a Mordell curve by looking at a pair of curves with unusually high rank.

Madeline Dawsey (University of Texas at Tyler)

Digital representations and special sequences

We explore non-standard binary and quaternary representations of integers. A short binary signed-digit (BSD) representation of an integer $n$ is a binary representation of $n$ with digits in $\{-1,0,1\}$ such that the leading two digits are neither $1,-1$ nor $-1,1$ . A hyperbinary representation of $n$ is a binary representation of $n$ with digit set $\{0,1,2\}$ . We show that the number of short BSD representations of a non-negative integer $n$ and the number of hyperbinary representations of $n-1$ are both equal to the $n$ -th term of the Stern sequence. We investigate similar properties of the number of balanced quaternary representations with digit set $\{-2,-1,0,1,2\}$ and hyperquaternary representations with digit set $\{0,1,2,3,4\}$ , which similarly satisfy a shifted identity and whose local maximum values coincide with the Fibonacci sequence.

Christian Ennis (Louisiana State University)

Convergence rates around $s = 1$ of the Riemann zeta function

The Riemann zeta function is classically studied in analytical number theory, with one major application being the prime number theorem. In the real sense, if $s = 1$ , we recover the harmonic series. However, if each term instead has varying powers $s_n$ which converge to 1, different convergence rates can allow the series to converge or diverge. We will discuss these rates using only elementary techniques from analysis.

Brian Grove (Louisiana State University)

Special values of modular L-functions: A hypergeometric perspective

L-functions attached to modular forms are fundamental objects in number theory that encode important arithmetic information, such as the distribution of ranks for quadratic twists of elliptic curves. I will discuss an explicit method to evaluate special values for a certain family of modular L-functions using input from hypergeometric functions over $\mathbb{C}$ .

Rajat Gupta (University of Maine)

Smallest parts of partitions into distinct parts

In this talk, we first review a classic theorem of Uchimura, which states that the difference between the sum of the smallest parts of the partitions of $n$ into an odd number of distinct parts and the corresponding sum for an even number of distinct parts is equal to the number of divisors of $n$ . Then we introduce the notion of $k$ -th smallest part of a partition of $n$ , $s_k(\pi)$ , and extend the Uchimura’s result. This is joint work with Noah Lebowitz-Lockard and Professor Joseph Vandehey.

Koustav Mondal (Louisiana State University)

Relating elliptic curve point-counting and coefficients of congruent theta functions

Sieving operators and V-operators are essential tools for exploring the properties of modular forms. We begin by defining these tools and outlining their key properties for our aid. Finally, we showcase an example where we establish the relationship between the Fourier coefficients of congruent theta series and the number of points on an elliptic curve.

Mohit Tripathi (Texas Tech University)

Splitting hypergeometric functions over roots of unity

In this talk, we will discuss the definition of hypergeometric functions over both the complex numbers and finite fields. Following that, we will prove a result that splits the hypergeometric function over finite fields at the $n$ -th roots of unity. We will then explore the applications of this theorem. This is joint work with Dermot McCarthy.