Generalizing Rasmussen's s-invariant, and applications

I will discuss a method to define Khovanov and Lee homology for links in connected sums of copies of S1 times S2.  From here we can define an s-invariant that gives genus bounds on oriented cobordisms between links.  I will discuss some applications to surfaces in certain 4-manifolds, including a proof that the s-invariant cannot detect exotic 4-balls coming from Gluck twists of the standard 4-ball.  If time allows, I will also discuss our new combinatorial proof of the slice Bennequin inequality in S1 times S2.  All of this is joint work with Ciprian Manolescu, Marco Marengon, and Sucharit Sarkar.