ABSTRACT:

This talk will describe some smoothness and function analytic properties of certain natural measures on path and loop spaces of compact Riemannian manifolds and compact Lie groups. The study of these questions is motivated, in part, by problems in physics relating to the path integral interpretation of quantum mechanics. I will briefly describe some (or at least one) such problems. Of course ``functional integration methods'' in the context of quantum field theory is responsible for a number of interesting mathematical conjectures. Although I may touch on one such conjecture, the talk will concentrate on what can be said in the more mathematically tractable setting of path and loop spaces of finite dimensional manifolds.