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

杨米尔斯理论、随机生长曲面和相关系统的数学基础

• 批准号: 2153654
• 负责人: Sourav Chatterjee
• 金额: $ 33万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153654&HistoricalAwards=false
• 关键词: Mathematical; Foundations; Yang; Mills; Theory

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.

该项目将研究概率理论的几个问题。头等舱涉及欧几里得阳米尔斯理论的构建。 Yang-Mills的理论是粒子物理标准模型的基础，这些模型尚未具有严格的数学基础。该项目旨在朝着为Yang-Mills理论赋予数学基础的目标朝着进步。第二类问题是关于随机表面的生长和融合到Kardar-Parisi-Zhang（KPZ）缩放限制的。假设KPZ方程是随机界面生长的规范模型（本质上是在自然界中发生的任何粗糙表面），但是除了少数情况外，这种说法通常仍然超出了严格的数学范围。该项目旨在取得进步，以更全面地了解KPZ增长。该项目还将在研究生级别提供研究培训机会。Yang-Mills项目将建立有关Yang-Mills热量方程的新结果。当初始数据是具有一定规律性的函数时，它已经知道如何构建Yang-Mills热方程的解决方案。当初始数据是随机分布时，该项目将尝试首次构建Yang-Mills热方程的解决方案。然后，这些解决方案将用于构建杨米尔斯理论的状态空间。 KPZ方程式上的项目提供了一种新的方式，可以通过查看本地而不是全球行为来查看与KPZ方程的性质，并表明可以在适当假设下进行任意扩展限制来证明这种行为。该奖项反映了NSF的法定任务，并通过使用基础的智力来评估了NSF的法定任务，并以智力的评估值得进行评估，并具有依据的评估。

项目成果

The Yang-Mills heat flow with random distributional initial data

• DOI: 10.1080/03605302.2023.2169937
• 发表时间: 2021-11
• 期刊: Communications in Partial Differential Equations
• 影响因子: 1.9
• 作者: Sky Cao;S. Chatterjee