Curved Wiener Space Analysis

本提案主要涉及三个主题。首先研究了黎曼流形上维纳测度的几种新的有限维几何近似的收敛性。二是将经典的“骨架”和“泰勒同构”定理推广到路径群和环路群。第三是对Malliavin的提升方法进行改进，以推导关于热核和相关狄利克雷形式的信息。前两个话题在一定程度上是由私家侦探发起的他试图理解量子化Yang-Mills场这一重要问题，而Yang-Mills场是粒子物理学“标准模型”的关键部分。（见Clay数学研究所关于量子化Yang-Mills场的问题。）第三个主题是尝试从椭圆型和准椭圆型热方程解的路径积分表示中提取有用的信息。P.I.希望通过修改标准的Bismut和Malliavin矢量场提升技术，找到新的准椭圆扩散梯度不等式。自20世纪40年代以来，费曼“路径积分”在量子物理学的描述中发挥了核心作用。虽然路径积分的数学基础在很多情况下仍然是脆弱的。这一提议的大部分内容都致力于路径积分的数学，这反过来可能会影响我们对基本粒子描述的理解。所要解决的问题的目的是为物理学家在描述基本粒子时使用的启发式表达式和计算提供精确的数学意义。除了具有基础重要性外，该项目还应阐明量子化过程中可能出现的“异常”，即从物理系统的经典力学描述到量子力学描述的过程。本文还提出了利用这种现象的扩散解释来开发新的方法来提取有关复杂几何体中热流的有用信息。这个建议有一个重要的研究生培训组成部分，因为一些问题将由私家侦探解决的学生。

This proposal is primarily concerned with three topics. The first is to study the convergence properties of certain new finite dimensional geometric approximations to Wiener measure on a Riemannian manifold. The second is to generalize the classical "Skeleton" and "Taylor isomorphisms" theorems to path and loop groups. The third is to find refinements to Malliavin's lifting method for deducing information about heat kernels and related Dirichlet forms. The first two topics are motivated, in part, by the P.I.'s attempt to understand the important problem of quantizing Yang-Mills fields which form a key part of the "standard model" of particle physics. (See the Clay Mathematics Institute problem pertaining to quantized Yang-Mills fields for a description.) The third topic is an attempt to extract useful information from the path integral representations for solutions to elliptic and hypoelliptic type heat equations. The P.I. hopes to find new gradient inequalities for hypoelliptic diffusions by modifying the standard Bismut and Malliavin vector-field lifting techniques.Since the 1940's, Feynman "path integrals" have played a central role in the description of quantum physics. Although highly studied, the mathematical footing of path integrals in many contexts is still tenuous at best. Much of this proposal is devoted to the mathematics of path integrals which in turn may impact our understanding of the description of elementary particles. The problems to be addressed are aimed at giving mathematically precise meaning to the heuristic expressions and computations which are used by physicists in the description of elementary particles. Besides being of foundational importance, this project should shed light on the "anomalies" which can appear in the quantization process, i.e. the process of going from a classical mechanical description to a quantum mechanical description of a physical system. It is also proposed to develop new methods for extracting useful information about the flow of heat in complicated geometrical bodies using the diffusion interpretation of this phenomenon. This proposal has a significant graduate training component as a number of the problems will be tackled by the P.I.'s students.