Lattice Gauge Theories, Importance Sampling, and Quantum Unique Ergodicity

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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.