Georgia Institute of Technology College of Sciences

Let $A = (a_{ij})$ be an $n \times n$ matrix with entries in a field $\mathbb{F}$ and let $f$ be a function defined on $\mathbb{F}$. The function naturally induces an entrywise transformation of $A$ via $f[A] := (f(a_{ij}))$. The study of such entrywise transforms that preserve various forms of matrix positivity has a rich and long history since the seminal work of Schoenberg. In this talk, I will discuss recent developments in the setting that the underlying field $\mathbb{F}$ is the real field, the complex field, and finite fields. I will also highlight some interesting connections between these problems with arithmetic combinatorics, finite geometry, and graph theory. Joint work with Dominique Guillot, Himanshu Gupta, and Prateek Kumar Vishwakarma.

The Heil-Ramanathan-Topiwala (HRT) conjecture is an open problem in time-frequency analysis. It asserts that any finite combination of time-frequency shifts of a non-zero function in $L^2(\mathbb{R})$ is linearly independent. Despite its simplicity, the conjecture remains unproven in full generality, with only specific cases resolved.
In this talk, I will discuss the HRT conjecture for a specific symmetric configuration of five points in the time-frequency plane, known as the $(3,2)$ configuration. Building upon restriction principles, we prove that for this specific setting, the Gabor system is linearly independent whenever the parameters satisfy certain rationality conditions (specifically, when one parameter is irrational and the other is rational). This result partially resolves the remaining open cases for such configurations. I will outline the proof methods, which involve an interplay of harmonic analysis and ergodic theory. This is joint work with Kasso A. Okoudjou.

Abstract: In recent years we have witnessed a symbiotic trend wherein LLMs are being combined with provers, solvers, and computer algebra systems, resulting in dramatic breakthroughs in AI for math. Following this trend, we have developed two lines of work in my research group. The first is the idea that "good" joint embeddings (JE) can dramatically improve the efficacy of LLM-based auto-formalization tools. We say that JEs are good if they respect the following invariant: semantically-equivalent formally-dissimilar objects (e.g., pairs of sematically-equivalent natural and formal language proofs) must be "close by" in the embedding space, and semantically inequivalent ones "far apart". We use such JE models as part of a successful RAG-based auto-formalization pipeline, demonstrating that such JEs are a critical AI-for-math technology. The second idea is Reinforcement Learning with Symbolic Feedback (RLSF), a class of techniques that addresses the LLM hallucination problem in contexts where we have access to rich symbolic feedback such math, physics, and code, demonstrating that they too are critical to the success of AI for math. 

Bio: Dr. Vijay Ganesh is a professor of computer science at Georgia Tech and the associate director of the Institute for Data Engineering and Science (IDEaS), also at Georgia Tech. Additionally, he is a co-founder and steering committee member of the Centre for Mathematical AI at the Fields Institute, and an AI Fellow at the BSIA in Waterloo, Canada. Prior to joining Georgia Tech in 2023, Vijay was a professor at the University of Waterloo in Canada from 2012 to 2023, a co-director of the Waterloo AI Institute from 2021 to 2023, and a research scientist at the Massachusetts Institute of Technology from 2007 to 2012. Vijay completed his PhD in computer science from Stanford University in 2007. 

Vijay's primary area of research is the theory and practice of SAT/SMT solvers, combinations of machine learning and automated reasoning, and their application in neurosymbolic AI, software engineering, security, mathematics, and physics. In this context he has led the development of many SAT/SMT solvers, most notably, STP, Z3str family of string solvers, Z3-alpha, MapleSAT, AlphaMapleSAT, and MathCheck. He also leads the development of several neurosymbolic AI tools aimed at mathematics, physics, and software engineering. On the theoretical side, he works on topics in mathematical logic and proof complexity. For his research, Vijay has won over 35 awards, honors, and medals, including an ACM Impact Paper Award at ISSTA 2019, ACM Test of Time Award at CCS 2016, and a Ten-Year Most Influential Paper citation at DATE 2008.

Partial identification provides an alternative to point identification: instead of pinning down a unique parameter estimate, the goal is to characterize a set guaranteed to contain the true parameter value. Many partial identification approaches take the form of linear optimization problems, which seek the "best- and worst-case scenarios" of a proposed model subject to the constraint that the model replicates correct observable information. However, such linear programs become intractable in settings with multivalued or continuous variables. This paper introduces a novel method to overcome this computational and statistical curse of cardinality: an entropy penalty transforms these potentially infinite-dimensional linear programs into general versions of multi-marginal Schrödinger bridges, enabling efficient approximation of their solutions. In the process, we establish novel statistical and mathematical properties of such multi-marginal Schrödinger bridges---including an analysis of the asymptotic distribution of entropic approximations to infinite-dimensional linear programs. We illustrate this approach by analyzing  instrumental variable models with continuous variables, a setting that has been out of reach for existing methods.

We present joint work with Artur Avila on delocalizing Schr\"odinger operators in arbitrary dimensions via arbitrarily small perturbations of the potential. As a consequence we obtain an analog of Simon's Wonderland Theorem for the case of dynamically defined potentials. We will discuss a mechanism based on the Feynman-Hellmann Theorem, whose infinite volume limit is instrumental in establishing delocalization in infinite volume. Furstenberg's correspondence principle then yields the desired delocalization statement in finite volume.

We study the SIRS process on sparse random graphs with power--law degree distributions.
A large physics literature reports numerical evidence for a positive epidemic threshold for SIRS with waning immunity on scale--free networks, suggesting a transition between short--lived and exponentially long--lived regimes.
In contrast, for the SIS/contact process on power--law graphs with exponent $\tau>3$, it is rigorously known that the critical value is $\lambda_c=0$ and that survival is exponentially long for every $\lambda>0$.
We show that, in a survival--time sense, the true threshold for SIRS on power--law random graphs with $\tau>3$ is also zero. Joint work with Debankur Mukherjee and Souvik Dhara. 

We give an overview of Teichmuller theory, the deformation theory of Riemann surfaces. The richness of the subject comes from all the perspectives one can take on Riemann surfaces: complex analytic for sure, but also Riemannian, topological, dynamical and algebraic.  In the past 40 years or so, interest has erupted in an extension of Teichmuller theory, here thought of as a component of the character variety of surface group representations into PSL(2,\R), to the study of the character variety of surface group representations into higher rank Lie groups (e.g. SL(n, \R)). We give a even breezy  discussion of that.  The first talk will begin with a segment that recalls scenes from the first overview in November.