In-Person and Online Talk 

*Please note the change in time*

We discuss the history of applying localization results in Lagrangian Floer homology to Heegaard Floer homology to obtain rank inequalities for various versions of the Heegaard Floer homology of branched double covers. We then use recent refinements on the symplectic side to establish an inequality between the dimensions of the Heegaard Floer homologies of any double cover between three-manifolds branched along a nulhomologous link. We discuss the relationship with the L-space conjecture and (time permitting) mention some other topological applications. This is joint work with R. Lipshitz and T. Lidman.