# Relative adjunction inequalities and their applications

I'll discuss ongoing joint work with Katherine Raoux that uses knot Floer homology to establish relative adjunction inequalities. These inequalities bound the Euler characteristics of properly embedded smooth cobordisms between links in the boundary of certain smooth 4-manifolds. The inequalities generalize the slice genus bound for the "tau" invariant studied by Ozsvath-Szabo and Rasmussen. I will use our inequalities to define concordance invariants of links, prove new results about contact structures, motivate a 4-dimensional interpretation of tightness, and to show that knots with simple Floer homology in lens spaces (or L-spaces) minimize rational slice genus amongst all curves in their homology class, upgrading a remarkable result of Ni and Wu pertaining to the rational Seifert genus.