Heat Kernels and Path Integrals

私家侦探将研究三个问题。第一个问题涉及到随机方法研究热核边界。我们的目标是找到独立的尺寸，因此适用于无限维热流的估计。 在亚椭圆热核的情况下，P.I.正在寻找精确的热核上限和下限第二个问题是研究无限维李群上热核测度的拟不变性和光滑性。私家侦探希望找到无限维次椭圆算子是次椭圆的例子。这些结果的证明将依赖于第一个问题的结果。第三个问题涉及到有限维近似的欧几里德费曼路径积分的理解。特别是，它建议进行此程序的超对称路径积分。这些结果和它们的推广应该有分支的Stolz-Teichner程序的几何化椭圆上同调。在20世纪40年代，费曼提出了一个美丽的，但高度启发性的理论，用于理解量子力学在经典力学中出现的更熟悉的概念。 费曼的启发式思想对拓扑学的数学理论产生了重大影响。（拓扑学是研究曲面和其他更一般的“流形”的基本形状属性。“）P.I.将在基本粒子物理学的背景下研究一些与费曼模型相关的问题。尽管已经研究了70多年，但这些理论的数学基础仍然没有得到很好的理解，许多有趣的问题仍然没有解决。例如，物理学家认为“夸克”应该被限制。然而，在目前的模型状态下，夸克禁闭的理论推导是遥不可及的。这个建议的具体目标是：1）使用概率技术来描述高维空间中热流的定性和定量行为，2）发展理解“超对称费曼路径积分”的方法。“希望在本资助期间开发的结果将适用于粒子物理模型和数学拓扑结构。这项建议包括一个培训部分，因为有些问题将交给研究生。

The P.I. will study three problems. The first problem involves the study of heat kernel bounds by stochastic methods. The goal is to find estimates which are independent of dimension and hence applicable to infinite dimensional heat flows. In the case of hypoelliptic heat kernels, the P.I. is looking for precise heat kernel upper and lower bounds. The second problem is to study the quasi-invariance and smoothness properties of heat kernel measures on infinite dimensional Lie groups. The P.I. hopes to find examples of infinite dimensional subelliptic operators which are hypoelliptic. The proof of these results will rely on the results of the first problem. The third problem relates to the understanding of Euclidean Feynman path integrals by finite dimensional approximations. In particular, it is proposed to carry this program out for super-symmetric path integrals. These results and their generalizations should have ramifications to the Stolz-Teichner program of geometrizing elliptic cohomologies. In the 1940's, Feynman proposed a beautiful, but highly heuristic, theory for understanding quantum mechanics in terms of the more familiar concepts arising in classical mechanics. Feynman's heuristic ideas have had a significant influence on the mathematical theory of topology. (Topology is the study of the basic shape properties of surfaces and other more general "manifolds.") The P.I. will investigate a number of problems related to Feynman's model in the context of elementary particle physics. Despite being studied for 70 plus years, the mathematical foundations of these types of theories are still not well understood and many interesting questions remain open. For example, physicists believe that "quarks" should be confined. However, with the current state of the models, the theoretical derivation of quark confinement is out of reach. The specific goals of this proposal are: 1) to use probabilistic techniques to describe qualitative and quantitative behavior of the flow of heat in high dimensional spaces, and 2) to develop methods for understanding the "super-symmetric Feynman path integral." It is hoped that the results developed during this grant period will be applicable to the particle physics models and to mathematical topology. This proposal includes a training component as some of the problems will be given to graduate students.