CAREER: Heat kernel measures in infinite dimensions

本研究计划统一为无限维椭圆和次椭圆热核测度的光滑性研究。这个建议的焦点-光滑性，准不变性和泰勒同构-是研究无限维测度的基本原则。虽然这些概念有其内在的意义，但人们可以通过物理应用来激发他们的研究。例如，在量子化的经典力学中，标准的做法是对某些无限维空间进行关于“无限维勒贝格测度”的积分。当然，这个测度是不存在的，合适的替代品是维纳测度，或者对于无限维的弯曲空间，热核测度。由于物理学家进行的积分计算通常涉及变量和微分的变化，热核测度的准不变性和更一般的光滑性结果在为这些形式计算提供数学基础方面至关重要。泰勒同构定理在物理应用中也很重要，因为它们涉及到量子化杨-米尔斯场的重要问题，而杨-米尔斯场是粒子物理学“标准模型”的关键部分。亚椭圆热核测度的研究在物理模型中出现的亚黎曼几何的研究中具有重要意义，是概率论研究与训练的综合项目。该研究计划的重点是研究所谓的？热核措施？在无限的维度上。考虑一个弯曲的表面（如地球），一个人在一个固定的点上施加一个单位的热量，然后离开，让热量传播。热核测度描述了在稍后的某个时间表面的子集中的热的比例的演化。由于热扩散的方式在很大程度上依赖于相关空间的几何和拓扑性质，热核测度反映了这些性质，对该测度的分析揭示了它所处的空间。热核测度是高斯测度（正态分布的高维类似物）的弯曲空间的自然推广，几十年来，它被认为是进行无限维分析的适当测度。无限维空间经常出现在物理模型中;特别是，无限维群出现在量子场论和量子力学中。因此，该研究计划有助于巩固量子物理研究的严格数学框架。许多应用程序都有有限维模型，本科生或刚开始研究生都可以使用。 因此，这个项目有一个重要的培训组成部分，包括在有限维的相关问题的研究涉及本科生。 该培训计划还包括指导研究生介绍数学推广，并促进女性研究人员在概率方面的保留和知名度。

This research project is unified as a study of smoothness properties of elliptic and hypoelliptic heat kernel measures in infinite dimensions. The focal points of this proposal - smoothness, quasi-invariance, and Taylor isomorphisms - are fundamental principles in the study of measures in infinite dimensions. Although these concepts have their own intrinsic interest, one may motivate their study by physical applications. For example, in quantized classical mechanics, it is standard practice to perform integration over certain infinite-dimensional spaces with respect to "infinite-dimensional Lebesgue measure." Of course, this measure does not exist, and the appropriate replacement is Wiener measure or, for an infinite-dimensional curved space, heat kernel measure. As the integral computations physicists perform routinely involve changes of variable and differentiation, quasi-invariance and more general smoothness results for heat kernel measures are critical in giving a mathematical foundation to these formal computations. Taylor isomorphism theorems are also significant to physical applications as they are related to the important problem of quantizing Yang-Mills fields which form a key part of the "standard model" of particle physics. The study of hypoelliptic heat kernel measures are important in the study of sub-Riemannian geometries that appear in physical models.This is an integrated project of research and training in probability. The research program focuses on the study of so-called ?heat kernel measures? in infinite dimensions. Consider a curved surface (like the Earth) to which one applies a unit of heat at a fixed point and then steps away and allows the heat to propagate. The heat kernel measure describes the evolution of the proportion of heat in subsets of the surface at some later time. As the manner in which heat diffuses relies heavily on the geometric and topological properties of the relevant space, the heat kernel measure reflects these properties and analysis of this measure reveals much about the space on which it lives. Heat kernel measures are the natural generalization to curved spaces of Gaussian measures (higher dimensional analogues of the normal distribution) which have for many decades been recognized as the appropriate measures to conduct infinite-dimensional analysis. Infinite-dimensional spaces often show up in physical models; in particular, infinite-dimensional groups appear in quantum field theory and quantum mechanics. Thus, this research program contributes to solidifying a rigorous mathematical framework for the study of quantum physics. Many of the applications have finite-dimensional models which are accessible to undergraduates or beginning graduate students. Thus, this project has a significant training component, including involving undergraduates in research on related problems in finite dimensions. The training program also includes mentoring of graduate students in an introduction to math outreach and promoting the retention and visibility of women researchers in probability.