Link Slicing and Casson towers

A link is slice if it bounds a collection of disjointly embedded locally flat discs in the 4-ball. A Casson tower is a 4-manifold with a fixed attaching circle, built from immersed thickened discs, with each stage of discs in the tower bounding double point loops of the discs in the previous stage. Instead of looking for an embedded slice disc for a link, one can instead try to find a Casson tower bounded by the link, which might be easier. By work of M. Freedman, a Casson tower of height 6 contains a locally flat embedded disc. We discuss various sharpenings of this result involving lower heights which enable us to give new examples of slice links. Joint with Jae Choon Cha.