Floer-theoretic filtrations on cohomology via C*-actions

In this talk, I will explain how one can obtain filtrations on ordinary and quantum cohomology of a symplectic manifold, using the Hamiltonian Floer homologies associated with a pseudo-holomorphic C*-action on it. This filtration is an invariant of the manifold together with a C*-action. It should be thought of as a Floer-theoretic analogue of Atyiah-Bott filtration, and it is computable in examples using certain Morse-Bott-Floer spectral sequences, as I will show in examples. Time permitting, I will discuss its comparison with P=W filtration on Higgs moduli spaces. This is a joint work with Alexander Ritter.