Loop and Path Space Analysis

9971036驱动程序本研究主要致力于研究黎曼流形的路空间和圈空间上与热核和Wiener测度相关的几何分析。P.I.将继续研究紧致黎曼流形上路径和循环空间上的路径积分公式的有限维逼近。这些近似将导致：1)Atiyah-Singer指标定理的启发式路径积分证明的严格解释；2)路和环空间上布朗运动的构造；3)环群上非平凡调和微分形式的构造；4)环空间上Dirac算子的构造。这些目标与主要研究人员正在进行的理解循环空间的指数定理和Hodge-de Rham定理的项目是一致的。本文还试图将L.Gross和P.I.的一个结果(与Segal Bargmann变换有关)从复有限维群推广到复循环群。在曲面和高维流形的数学研究中，自然会出现环的空间。例如，如果曲面上有一个环，在不撕裂该环的情况下无法将其变形到某一点，则该曲面必须包含一个“孔”。(请考虑甜甜圈的表面与球体的表面。)物理学家对自然界基本粒子的弦理论描述也使用了环路或“弦”。弦理论的量子力学处理导致了与P.I.提出的研究问题类似的问题。这个建议致力于回答关于有限维曲面和此类曲面上的回路上的量子力学(Feynman)路径积分公式的基本数学问题。作为这项研究的副产品，P.I.希望为所谓的Atiyah-Singer指数定理提出严谨的、非常漂亮的物理论点。这个指数定理可以说是本世纪最引人注目的数学发展之一，它在数学和物理学中都有深远的影响。

9971036 Driver This research is primarily devoted to the study of the geometric analysis associated to heat kernel and Wiener measures on path and loop spaces of Riemannian manifolds. The P.I. will continue his study of finite dimensional approximations to path integral formulas on path and loop spaces over compact Riemannian manifolds. It is hoped that these approximations will lead to: 1) a rigorous interpretation of the heuristic path integral proofs of the Atiyah--Singer index theorem, 2) construction of Brownian motions on path and loop spaces, 3) construction of non-trivial harmonic differential forms on loop groups, and 4) construction of Dirac operators on loop spaces. These goals are consistent with the principal investigators ongoing project to understand index theorems and the Hodge -- de Rham theorem for loop spaces. It is also proposed to try to generalize a result (related to the Segal Bargmann transform) of L. Gross and the P.I. from complex finite dimensional groups to complex loop groups. Spaces of loops naturally arise in the mathematical study of surfaces and of higher dimensional manifolds. For example, if there is a loop on a surface that can not be deformed to a point without tearing the loop apart, then the surface must contain a "hole." (Think of the surface of a doughnut versus the surface of a sphere.) The physicists' string theory description of the fundamental particles of nature also use loops or "strings." The quantum mechanical treatment of string theory leads to problems similar to those proposed for study by the P.I. This proposal is devoted to answering basic mathematical questions about the (Feynman) path integral formulation of quantum mechanics on both finite dimensional curved surfaces and on the loops on such surfaces. As a by-product of this research, the P.I. hopes to make rigorous the very beautiful physics argument for the so-called Atiyah-Singer Index theorem. This index theorem is arguably one of the most striking mathematical developments of this century and it has had far reaching implications in both mathematics and in physics.