Equivariant localization and Atiyah-Segal completion for Hochschild and cyclic homology

There is a close relationship between derived loop spaces, a geometric object, and Hochschild homology, a categorical invariant, made possible by derived algebraic geometry, thus allowing for both intuitive insights and new computational tools.  In the case of a quotient stack, we discuss a "Jordan decomposition" of loops which is made precise by an equivariant localization result.  We also discuss an Atiyah-Segal completion theorem which relates completed periodic cyclic homology to Betti cohomology.