CAREER: Properties of Solutions to Singular Stochastic Partial Differential Equations from Quantum Field Theory

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

This mathematics research project focuses on studying the small-scale behavior of some quantum field theory models in physics. The research aims to produce an accurate description and understanding of the ultraviolet divergence phenomenon. In the study of the standard model of elementary particles, for example, physicists often use Monte-Carlo simulation; the research carried out in this project aims to reveal why discrete simulation can accurately predict the behavior of a continuous quantum field, how quickly dynamical algorithms will converge to the quantity being observed, and to what degree a homogenized approximation can effectively describe the macroscopic nature of a many-body problem. The project also includes two educational components, focusing on rejuvenation of the Graduate Student Probability Conference (GSPC), which is entirely run by graduate students primarily from the US, and a Stochastic Partial Differential Equations (SPDE) education program that is vertically integrated across the full spectrum of academic research. The GSPC and the SPDE education program seek sustained long-term benefit to the future generations of the probability community. The project focuses on studying the properties of the solutions of some singular stochastic partial differential equations. In recent years, there has been rapid progress in the construction of the solutions of these equations, and the project aims to study the properties of the solutions that have been constructed. Specifically, the investigator will study the universal limits of the stochastic Yang-Mills equation and explore the connections between stochastic partial differential equations, large-N problems in quantum field theory, and the theory of mean field limits. Recently developed theory of solutions will play a central role. The PI aims to develop rigorous proofs for results that make practical predictions. These studies are anticipated to result in profound connections with physics and even to influence other natural sciences. 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.