Research over the last fifteen years has uncovered a raft of inequalities between the Floer homology invariants of a 3-manifold and of certain of its covers. We will recall some of these results, the philosophy behind them, and some of their implications and connections with classical and modern conjectures. We will then describe a new technique for proving such results, due to Tim Large, and a recent application of it, in joint work with Kristen Hendricks and Tye Lidman. Speaker(s): Robert Lipshitz (University of Oregon)