Curved Wiener Space Analysis

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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.