Mathematical Sciences: Loop Space Analysis

驾驶员将在路径上学习微积分 和紧黎曼流形的圈空间。 长期 目标是开发Malliavin演算，Hodge和索引类型 关于曲线上的路和环的Wiener空间的定理 流形 具体问题包括路径构建和 随机发展的圈空间值布朗运动 使用双参数随机微积分，并调查 不同路径和回路的log-Sobolev不等式的存在性 空间措施。 该项目涉及循环空间。在特殊情况下， 二维流形，循环空间就是 曲面上的所有连续闭合路径， 不动点 一个关键的想法是构建一个衡量这些 这样我们就可以讨论一组路径的“体积”， 从而在环路空间上积分。 这是非常重要的 理解循环空间的拓扑结构， 物理学中的弦理论

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.