Mathematical Foundations for Yang-Mills Theory, Randomly Growing Surfaces, and Related Systems

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This project will study several questions in probability theory. The first class concerns the construction of Euclidean Yang-Mills theories. Yang-Mills theories are the building blocks of the Standard Model of particle physics, which do not yet have a rigorous mathematical foundation. The project aims to make progress towards the goal of giving a mathematical foundation to Yang-Mills theories. The second class of questions is about the growth of random surfaces and convergence to Kardar-Parisi-Zhang (KPZ) scaling limits. The KPZ equation is hypothesized to be the canonical model for the growth of random interfaces (essentially, any rough surface occurring in nature), but other than in a handful of cases, such claims generally remain out of the reach of rigorous mathematics. The project aims to make progress towards a more comprehensive understanding of KPZ growth. The project will also provide research training opportunities at graduate level.The Yang-Mills project will establish new results about the Yang-Mills heat equation. It is known how to construct solutions to the Yang-Mills heat equation when the initial data is a function with some regularity. This project will attempt to construct, for the first time, a solution of the Yang-Mills heat equation when the initial data is a random distribution. These solutions will then be used to construct state spaces for Yang-Mills theories. The project on the KPZ equation provides a new way to look at the nature of convergence to the KPZ equation, by looking at local, rather than global, behavior, and showing that such behavior can be proven for arbitrary scaling limits under suitable assumptions.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.