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。第一类是关于欧几里得杨-米尔斯理论的构造。杨-米尔斯理论是粒子物理学标准模型的基石，它还没有严格的数学基础。该项目的目的是朝着杨米尔斯理论的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和收敛到Kardar-Parisi-Zhang（KPZ）标度极限。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）生长的规范模型，但除了少数情况外，这种说法通常仍然超出严格数学的范围。该项目旨在更全面地了解KPZ的增长。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。已知当初始数据是具有某种正则性的函数时，如何构造杨-米尔斯热方程的解。本项目将首次尝试构造初始数据为随机分布时杨-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。KPZ方程的项目提供了一种新的方式来看待收敛到KPZ方程的性质，通过查看局部，而不是全局，行为，并表明这种行为可以证明在适当的假设下任意缩放限制。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.