Contact Dehn surgery, symplectic fillings, and Property P for knots

These are notes of a talk given at the Mathematische Arbeitstagung 2005 in Bonn. Following ideas of Özbağcı–Stipsicz, a proof based on contact Dehn surgery is given of Eliashbergʹs concave filling theorem for contact 3-manifolds. The role of the theorem in the Kronheimer–Mrowka proof of Property P for nontrivial knots is sketched.