Floer homology and Dehn fillings
Peter Ozsváth, Columbia University

I will discuss joint work with Peter Kronheimer, Tom Mrowka, and Zoltan Szabo, in which we use methods from gauge theory to verify a conjecture of Gordon, according to which if p/q Dehn surgery on a knot is the lens space L(p,q), then the knot is the unknot. The key technical device is a surgery long exact sequence for Seiberg-Witten monopole Floer homology. There are other applications of these techniques to problems of lens space surgeries, and also to the non-existence of taut foliations over certain three-manifolds.