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

这个项目将研究概率论中的几个问题。第一节课是关于欧几里德杨-米尔斯理论的构建。杨-米尔斯理论是粒子物理标准模型的基石，该模型还没有严格的数学基础。该项目旨在朝着为杨-米尔斯理论提供数学基础的目标取得进展。第二类问题是关于随机曲面的增长性和收敛到Kardar-Parisi-Zhang(KPZ)尺度极限。KPZ方程被假设为随机界面(本质上是自然界中出现的任何粗糙表面)增长的正则模型，但除了少数情况外，这种说法通常都超出了严格的数学能力范围。该项目旨在朝着更全面地了解KPZ增长的方向取得进展。该项目还将提供研究生水平的研究培训机会。杨-米尔斯项目将建立关于杨-米尔斯热方程的新结果。当初始数据是具有一定规律性的函数时，如何构造杨-米尔斯热方程的解是已知的。本项目将首次尝试在初始数据为随机分布的情况下构造杨-米尔斯热方程的解。这些解将被用来构造杨-米尔斯理论的状态空间。关于KPZ方程的项目提供了一种新的方法来看待收敛到KPZ方程的性质，通过观察局部行为而不是全局行为，并表明在适当的假设下，这种行为可以被证明为任意比例限制。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.