Equivariant Structures in Symplectic Floer Homotopy

I will discuss a construction of a genuine Z/kZ equivariant homotopy type associated to the k-th iterate of a Hamiltonian diffeomorphism. Making sense of this construction requires an extension of the Cohen-Jones-Segal construction of Floer homotopy types to the setting of virtually smooth flow categories, in which the morphism spaces (i.e. moduli spaces of flow lines) are no longer smooth manifolds with corners, but instead admit a compatible system of Kuranishi charts.  This construction elucidates obstruction-bundle computations in equivariant Morse theory, and will be motivated by connections to noncommutative Hodge theory and to string topology.