Lattice Gauge Theories, Importance Sampling, and Quantum Unique Ergodicity

本研究项目研究概率论中的三类问题。其中一类涉及格点规范理论，它是量子场论的离散近似。量子场论是现代理解粒子物理的核心，但还没有坚实的数学基础，对量子场论的正确数学理解长期以来不仅是数学家的目标，也是理论物理学家的目标。这个项目旨在阐明这一领域的一些基本数学问题。第二类问题涉及重要性抽样的理论性质以及这些性质的各种应用。重要性抽样是发展计算统计学坚实基础的核心；这项工作的结果预计将在使用科学计算的数学以外的广泛研究领域中有用，包括计算机科学、物理、化学、计算生物学和各种工程学科。第三类问题集中在建立动力系统性质的概率技术上。正在研究的问题连接了几个数学领域，包括数论、微局部分析和偏微分方程式。该项目包括对研究生进行概率论及其应用方面的培训。该项目将研究概率中的几个问题。其中一类问题涉及格点规范理论中威尔逊环期望的计算，它在物理学中有重要的应用。该项目的目的是在这个主题中提供一个罕见的严格结果，为著名的格点规范理论的1/N展开提供第一个适当的数学证明。第二类问题涉及重要抽样的理论性质以及这些性质的各种应用。该项目旨在解决在任何给定环境下确定重要抽样良好性能所需的最小样本量的数学问题。最后，第三类问题围绕着证明狄利克雷拉普拉斯的小扰动是量子唯一遍历的概率技术。

This research project investigates three classes of questions in probability theories. One class concerns lattice gauge theories, which are discrete approximations of quantum field theories. Quantum field theories are central to the modern understanding of particle physics but do not yet have a firm mathematical foundation, and a proper mathematical understanding of quantum field theories has long been a goal of not only mathematicians, but also of theoretical physicists. This project aims to shed light on some fundamental mathematical questions in this area. A second class of questions involves theoretical properties of importance sampling and various applications of these properties. Importance sampling is central to development of a strong foundation for computational statistics; the results of this work are anticipated to be useful in a wide variety of research areas outside mathematics in which scientific computing is used, including computer science, physics, chemistry, computational biology, and a variety of engineering disciplines. The third class of questions centers on probabilistic techniques for establishing properties of dynamical systems. The questions under study connect several areas of mathematics, including number theory, microlocal analysis, and partial differential equations. The project includes training of graduate students in probability theory and its applications. The project will study several questions in probability. One class of problems concerns the evaluation of Wilson loop expectations in lattice gauge theories, which has important applications in physics. The project aims to provide a rare rigorous result in this topic, giving the first proper mathematical justification for the famous 1/N expansion of lattice gauge theories. A second class of problems involves theoretical properties of importance sampling and various applications of these properties. The project aims to solve the mathematical problem of determining the minimum sample size required for good performance of importance sampling in any given setting. Lastly, a third class of problems centers around probabilistic techniques for proving that small perturbations of Dirichlet Laplacians are quantum unique ergodic.