Vanderbilt University Number Theory Seminar

Welcome to the seminar page for the Vanderbilt number theory group. We have a vibrant group, currently consisting of Jayashree Kalita, Wanlin Li, Eleanor McSpirit, Andreas Mono, Larry Rolen, and Mohao Yi. If you would like to be added to the seminar email list, let Larry Rolen know by emailing larry.rolen@vanderbilt.edu.


	Upcoming Talks: Date and location: April 14, 2026, 10:00 AM (Central Time), SC 1404 Speaker: William Duke (University of California, Los Angeles) Title: New directions in the geometry of numbers Abstract: The geometry of numbers is a relatively old and highly developed area of number theory. One basic problem here is to estimate the minimal absolute value of a real homogeneous polynomial $P$ on $\R^n$ when evaluated at non-zero points of $\Z^n$. Typically, such an estimate is given in terms of $\mathrm{SL}(n,\R)$ invariants of the polynomial. A major refinement is to understand the {\it spectrum}, the set of minima of all of those polynomials in the $\mathrm{SL}(n,\R)$-orbit of $P$. In his recent lectures, Sarnak explains how these problems are part of a much more general program that studies the “bass note” spectra of certain linear differential operators. A different kind of generalization occurs when we replace the lattice $\Z^n$ by the set of integer points on an affine homogeneous variety and seek to minimize the absolute value of a polynomial restricted to these integers. In one of the simplest interesting cases the integer points are those on an (affine) quadric surface. In this talk, after going over the background, I will describe analogues of some well-known results about the minima of quadratic and cubic forms and their spectra, when we restrict them to the integer points on this quadric surface. Date and location: April 16, 2026, 12:30 PM (Central Time), SC 1404 Speaker: Jayashree Kalita (Vanderbilt University) Title: From a Conjecture of Andrews to Almost Alternating Sign Patterns Abstract: Computer experiments led Andrews, in 1986, to conjecture striking sign patterns and growth phenomena for the coefficients of five partition-theoretic q-series from the Ramanujan’s Lost Notebook. The first of these functions, the now-famous series \[ \sigma(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q;q)_n} \] 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 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 \[ v_1(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}. \] Jointly with Kundu, Storzer and 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.

Upcoming Talks:

Date and location: April 14, 2026, 10:00 AM (Central Time), SC 1404

Speaker: William Duke (University of California, Los Angeles)

Title: New directions in the geometry of numbers

Abstract: The geometry of numbers is a relatively old and highly developed area of number theory. One basic problem here is to estimate the minimal absolute value of a real homogeneous polynomial $P$ on $\R^n$ when evaluated at non-zero points of $\Z^n$. Typically, such an estimate is given in terms of $\mathrm{SL}(n,\R)$ invariants of the polynomial. A major refinement is to understand the {\it spectrum}, the set of minima of all of those polynomials in the $\mathrm{SL}(n,\R)$-orbit of $P$.

In his recent lectures, Sarnak explains how these problems are part of a much more general program that studies the “bass note” spectra of certain linear differential operators. A different kind of generalization occurs when we replace the lattice $\Z^n$ by the set of integer points on an affine homogeneous variety and seek to minimize the absolute value of a polynomial restricted to these integers.

In one of the simplest interesting cases the integer points are those on an (affine) quadric surface. In this talk, after going over the background, I will describe analogues of some well-known results about the minima of quadratic and cubic forms and their spectra, when we restrict them to the integer points on this quadric surface.

Date and location: April 16, 2026, 12:30 PM (Central Time), SC 1404

Speaker: Jayashree Kalita (Vanderbilt University)

Title: From a Conjecture of Andrews to Almost Alternating Sign Patterns

Abstract: Computer experiments led Andrews, in 1986, to conjecture striking sign patterns and growth phenomena for the coefficients of five partition-theoretic q-series from the Ramanujan’s Lost Notebook. The first of these functions, the now-famous series
\[
\sigma(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q;q)_n}
\]
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 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
\[
v_1(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}.
\]
Jointly with Kundu, Storzer and 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.