Georgia Institute of Technology College of Sciences

The Birkhoff Ergodic Theorem describes typical behaviors and averaged quantities with respect to an invariant measure. In this talk, I will focus on "observable" events, equating observability with positive Lebesgue measure. From this observational viewpoint, "typical" means typical with respect to Lebesgue measure. This leads immediately to issues for attractors, where all invariant measures are singular. I will present highlights of developments in smooth ergodic theory that address these questions. The theory of physical and SRB measures applies to dynamical systems that are deterministic as well as random, in finite and infinite dimensions (where observability has to be interpreted differently). This body of ideas argue in favor of convergence of ergodic averages for typical orbits. But the picture is a little more complicated: In the last part of the talk, I will discuss some recent work that shows that in many natural settings (e.g. reaction networks), it is also typical for ergodic averages 
to fluctuate in perpetuity due to heteroclinic-like behavior.

We discuss recent progress on the study of a range of problems relating to homomorphism counts in graphs. Perhaps the most celebrated such problem is Sidorenko's conjecture, which says that the number of copies of any fixed bipartite graph in another graph of given density is asymptotically minimised by the random graph. As well as talking about some of the recent results on this conjecture, we will touch on the positive graph conjecture and the study of norming graphs. If time permits, we will also say a little about similar problems in an arithmetic context.

We introduce proximal optimal transport divergences that provide a unifying variational framework interpolating between classical f-divergences and Wasserstein metrics. From a gradient-flow perspective, these divergences generate stable and robust dynamics in probability space, enabling the learning of distributions with singular structure, including strange attractors, extreme events, and low-dimensional manifolds, with provable guarantees in sample size.

We illustrate how this mathematical structure leads naturally to generative particle flows for reconstructing nonlinear cellular dynamics from snapshot single-cell RNA sequencing data,including real patient datasets, highlighting the role of proximal regularization in stabilizing learning and inference in high dimensions.

Bio: Markos Katsoulakis is a Professor of Applied Mathematics and an Adjunct Professor of Chemical & Biomolecular Engineering at UMass Amherst,  whose research lies at the interface of PDEs, uncertainty quantification, scientific machine learning, and information theory. He serves on the editorial boards of the SIAM/ASA Journal on Uncertainty Quantification, the SIAM Journal on Scientific Computing, and the SIAM Mathematical Modeling and Computation book series. He received his Ph.D. in Applied Mathematics from Brown University and his B.Sc. from the University of Athens. His work has been supported by AFOSR, DARPA, NSF, DOE, and the ERC.

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.