Curved Wiener Space Analysis

弯曲维纳空间分析

• 批准号: 0504608
• 负责人: Bruce Driver
• 金额: $ 17万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504608&HistoricalAwards=false
• 关键词: Curved; Wiener; Space; Analysis

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.

这项提议主要涉及三个主题。第一个是研究黎曼流形上某些新的有限维几何逼近Wiener测度的收敛性质。二是将经典的“骨架”和“泰勒同构”定理推广到路群和环群。第三个是对Malliavin的提升方法进行改进，从而推导出关于热核和相关的Dirichlet形式的信息。前两个主题的部分动机是，P.I.S试图理解形成粒子物理“标准模型”的关键部分的杨-米尔场的量子化这一重要问题。(有关说明，请参阅克莱数学研究所关于量子化杨-米尔斯场的问题。)第三个主题是尝试从椭圆型和亚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矢量场提升技术来找到亚椭圆扩散的新的梯度不等式。自1940年S以来，费曼的路径积分在量子物理的描述中发挥了核心作用。尽管人们对路径积分进行了大量的研究，但在许多情况下，路径积分的数学基础充其量仍然是薄弱的。这一建议的大部分内容都致力于路径积分的数学计算，这反过来可能会影响我们对基本粒子描述的理解。要解决的问题旨在赋予物理学家在描述基本粒子时使用的启发式和计算以数学上的准确含义。除了具有基础性的重要性外，这个项目还应该阐明在量子化过程中可能出现的“反常”，即从一个物理系统的经典力学描述到量子力学描述的过程。还建议发展新的方法，利用对这一现象的扩散解释来提取关于复杂几何体中热流动的有用信息。这项提议有一个重要的研究生培训部分，因为许多问题将由P.I.S的学生来解决。