(Joint work with J. Rognes, Oslo)

In this talk I will explain the construction and properties of a certain chain complex H(t) associated to a given fixed twisting cochain t. Since this construction generalizes that of both the Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra, we call H(t) the Hochschild complex of t.

I'll give conditions under which H(t) admits power maps extending the usual power maps on a Hopf algebra. In particular, it turns out that both the Hochschild complex of any cocommutative Hopf algebra and the coHochschild complex of the normalized chain complex of a double suspension admit power maps, which are algebraic models for the topological power maps on free loop spaces. This algebraic model of the topological power map is a crucial element of the construction of our model for computing spectrum homology of topological cyclic homology of spaces.

In this talk I will outline a few of the important roles   that free loop spaces play in topology and mathematical physics.  In  particular, I will talk about enumeration of geodesics on manifolds, as well as about the relationship between Hochschild homology and free loop spaces.  Moreover I will sketch how a free loop space gives rise to a homological conformal field theory, via string topology.

Two fundamental finiteness conditions in group theory are the properties of being finitely generated and of finitely presented. More general finiteness conditions using geometric and homological properties of groups were introduced by C.T.C. Wall. Until recently, it was unknown whether the homological conditions implied the geometric conditions. M. Bestvina and N. Brady showed that this is not the case. They constructed counterexamples using right angled Artin groups and a piecewise linear version of the Morse theory. Purpose of this talk is to introduce these groups and see some applications.