Traces and Characteristic Classes in Infinite Dimensions

This paper surveys topological results obtained from characteristic classes built from the two types of traces on the algebra of pseudodifferential operators of nonpositive order. The main results are the construction of a universal $\widehat{A}$-polynomial and Chern character that control the $S^1$-index theorem for all circle actions on a fixed vector bundle over a manifold, and $\mathrm{\mid }{\pi }_1(\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M^5))\mathrm{\mid }=\mathrm{\infty }$, for $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M^5)$ the diffeomorphism group of circle bundles $M^5$ with large first Chern class over projective algebraic Kaehler surfaces.