Back in 2004, Rasmussen extracted a numerical invariant from Khovanov-Lee homology, and used it to give a new proof of Milnor's conjecture about the slice genus of torus knots. In this talk, I will describe a generalization of Rasmussen's invariant to null-homologous links in connected sums of ^1 \times ^2. For certain links in ^1 \times ^2, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of null-homologous surfaces with boundary in the following four-manifolds: B^2 \times ^2, ^1 \times B^3, CP^2, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from B^4 by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard. This is based on joint work with Marco Marengon, Sucharit Sarkar, and Mike Willis.