CAREER: Heat kernel measures in infinite dimensions

本研究项目统一为无限维椭圆和亚椭圆热核测度的光滑性研究。这一建议的重点--光滑性、准不变性和泰勒同构--是研究无限维度量的基本原理。虽然这些概念有其内在的兴趣，但人们可以通过物理应用来激励他们的学习。例如，在量子化的经典力学中，标准的做法是在某些无限维空间上对“无限维勒贝格测度”进行积分。当然，这种度量是不存在的，适当的替代是Wiener度量，或者，对于无限维曲线空间，热核度量。由于物理学家进行的积分计算通常涉及变量的变化和微分，所以热核测量的准不变性和更一般的光滑性结果对于为这些形式计算提供数学基础至关重要。泰勒同构定理对物理应用也很重要，因为它们涉及到对杨-米尔场进行量子化的重要问题，这是粒子物理“标准模型”的关键部分。亚椭圆热核度量的研究在物理模型中出现的亚黎曼几何的研究中占有重要地位，这是一个概率研究和训练的综合项目。该研究计划的重点是研究所谓的热核措施？在无限的维度中。考虑一个曲面(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。热核度量描述了表面子集中的热比例在以后的某个时间内的演变。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，所以热核度量反映了这些性质，对该度量的分析揭示了它所在的空间的许多信息。热核度量是高斯度量(正态分布的高维类似)在曲线空间的自然推广，几十年来一直被认为是进行无限维分析的合适度量。无限维空间经常出现在物理模型中；特别是在量子场论和量子力学中出现了无限维群。因此，这一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，本科生或研究生初学者都可以使用。因此，这个项目有一个重要的培训部分，包括让本科生参与有限维度的相关问题的研究。该培训计划还包括在数学推广入门课程中对研究生进行指导，并促进女性研究人员在概率学方面的留住和可见度。

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.