A theorem by J.D.S. Jones from 1987 identifies the cohomology of the free loop space of a simply connected space with the Hochschild homology of the singular cochain algebra of this space. There are very strong relations between the Floer homology of cotangent bundles in symplectic geometry and the homology of free loop spaces of closed manifolds. In the light of these connections, one wants to have a geometric and Morse-theoretic identification of free loop space cohomology and the Hochschild homology of Morse cochain algebras in order to establish relations between Floer homology and Hochschild homology. After describing the underlying Morse-theoretic constructions and especially the Hochschild homology of Morse cochains, I will sketch a purely Morse-theoretic version of Jones' map and discuss its most important properties.
If there is an extension of the first talk then I will outline a purely Morse-theoretic proof of Jones' theorem on free loop space homology and Hochschild homology. I will further discuss compatibility results with product structures like the Chas-Sullivan loop product and give explicit Morse-theoretic descriptions of products in Hochschild cohomology in terms of gradient flow trees.