CAREER: Geometric Foundations of Stochastic Dynamics: From Critical Phenomena to Quantum and Algorithmic Systems

The geometry of random systems, from the fractal structure of critical phase transitions to the curvature of quantum spacetime, lies at the heart of some of the deepest open problems in modern mathematics and physics. Yet, despite remarkable progress in probability theory, conformal field theory, optimal transport, and machine learning, the geometric mechanisms governing these systems remain poorly understood, limiting both theoretical insight and practical application. In response, this project pursues four synergistic research directions: (1) developing novel coarse graining techniques to unveil fractal structures in critical stochastic systems such as the two dimensional stochastic heat flow and directed random polymers; (2) establishing a probabilistic foundation for the symplectic geometry of quantum Teichmüller space through semiclassical analysis of Liouville quantum gravity, targeting landmark conjectures including the Nekrasov, Rosly and Shatashvili conjecture and the analytic Langlands correspondence; (3) formulating entropy based selection principles in optimal transport on Riemannian and Lorentzian manifolds, with implications for understanding radiation entropy near gravitational singularities; and (4) building a rigorous, unified theory for the convergence of interacting particle dynamics underlying modern generative AI algorithms. Efforts to broaden the impact of this work include mentoring a broad cohort of students across multiple institutions, developing new graduate courses at the intersection of probability and machine learning, and co organizing research programs and workshops at national mathematics institutes. The project investigates the interplay between randomness, geometry, and dynamics across multiple scales and disciplines. The overarching objective is to identify the geometric structures, including fractal peak sets, symplectic coordinates on moduli spaces, entropy trapping along transport rays, and fluctuation limits of particle systems, that govern the behavior of complex stochastic and algorithmic systems. To this end, the investigator proposes leveraging techniques from renormalization theory, Gaussian multiplicative chaos, Jacobi field analysis in curved geometry, and martingale methods for stochastic PDEs. This approach seeks to transform foundational questions in mathematical physics and AI, from characterizing multifractality in critical models to guaranteeing convergence of gradient based generative algorithms, into a unified, geometrically grounded probabilistic framework that is both analytically rigorous and broadly applicable. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2542899 | Program: 01003031DB NSF RESEARCH & RELATED ACTIVIT,01002930DB NSF RESEARCH & RELATED ACTIVIT,01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Promit Ghosal | Institution: University of Chicago, CHICAGO, IL | Award Amount: $257,479 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2542899 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2542899.html