Floer Homology with DG Coefficients. Applications to Cotangent Bundles

Given a path-connected topological space X, a differential graded (DG) local system (or derived local system) is a module over the DGA of chains on the based loop space of X. I will explain how to define in the symplectically aspherical case Hamiltonian Floer homology with coefficients in a DG local system, how this homology fits into a filtered homological toolbox, and will present a number of dynamical applications to cotangent bundles. This is joint work with Jean-François Barraud, Mihai Damian and Vincent Humilière. The construction of Floer homology with enriched coefficients was originally discovered by Barraud-Cornea, and it was revisited over the years in different settings by Abouzaid, Charette, Zhou, and Rezchikov.