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 $\hat A$-polynomial and Chern character that control the $S^1$-index theorem for all circle actions on a fixed vector bundle over a manifold, and $|\pi_1({\rm Diff}(M^5))| = \infty$, for ${\rm Diff}(M^5)$ the diffeomorphism group of circle bundles $M^5$ with large first Chern class over projective algebraic Kaehler surfaces.Comment: Parts of Section 2.3 are not correct. This is discussed in T. McCauley, "S^1-Equivariant Chern-Weil Constructions on Loop Spaces," arXiv:1507.0862

Universal Deformation Formulae, Symplectic Lie groups and Symmetric Spaces

We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of a class of solvable Lie groups. We also study compatible co-products by generalizing the notion of smash product in the context of Hopf algebras

Universal Deformation Formulae for Three-Dimensional Solvable Lie groups

We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of every three-dimensional solvable Lie group. We also study compatible co-products by generalizing the notion of smash product in the context of Hopf algebras. We investigate in particular the dressing action of the `book' group on SU(2)