Abstract:Given a closed four-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X - B^4, with boundary a knot K. We give several methods to produce bounds on the genus of such surfaces in a fixed homology class. Our techniques include relative adjunction inequalities and the 10/8 + 4 theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold. We give an example showing that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds. Further, we give examples of knots that are topologically but not smoothly H-slice in some indefinite 4-manifolds. This is joint work with Marco Marengon and Lisa Piccirillo.