Mathematical Sciences: Loop Space Analysis

DIVER将学习紧致黎曼流形的路径空间和循环空间上的微积分。长期目标是发展曲线流形上路径和环的Wiener空间的Malliavin微积分、Hodge和指标型定理。具体问题包括利用双参数随机演算通过随机发展来构造路径和环空间值布朗运动，以及研究不同路径和环路空间度量下的LOG-Sobolev不等式的存在性。该项目涉及循环空间。在二维流形的特殊情况下，环空间就是曲面上从某个固定点开始的所有连续闭路的族。一个关键的想法是在这些路径上构造一个度量，这样人们就可以谈论一组路径的“体积”，从而在循环空间上进行积分。这对于理解循环空间的拓扑很重要，并对物理学中的弦理论有一定的影响。

Driver will study differential and integral calculus on path and loop spaces of compact Riemannian manifolds. The long term goal is to develop Malliavin calculus, Hodge and index type theorems for Wiener spaces of paths and loops on curved manifolds. The specific problems include constructing path and loop space valued Brownian motions by stochastic development using two-parameter stochastic calculus, and investigating the existence of log-Sobolev inequalities for different path and loop space measures. The project involves loop spaces. In the special case of two dimensional manifold, the loop space is just the family of all continuous closed paths on a surface which start at some fixed point. One key idea is to construct a measure on these paths so that one can talk about the "volume" of a set of paths and thus integrate over the loop space. This is important for understanding the topology of loop spaces and has implications for string theory in physics.