Mathematical Sciences: Loop Space Analysis

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Professor Driver will study differential and integral analysis on loop spaces. A primary goal is to prove an infinite dimensional version of Hodge's theorem for loop spaces. The notion of harmonic forms will be determined by an Ornstein-Uhlenbeck Lapacian whose exact definition will depend on the measure on the loop space. An important theme in this work is to prove a logarithmic Sobolev inequality for the resulting Dirichlet form. This project involves loop spaces. In the special case of a 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.