Waldhausen’s theorem fifty years on: proofs and generalizations.

In 1968 Waldhausen gave a beautiful proof that the three sphere has a unique (up to isotopy) Heegaard decomposition into two genus g handlebodies. I’ll show you Waldhausen’s proof and sketch one or two others. We'll then proceed to generalizations to other 3-manifolds, and parametric versions, e.g. the Powell conjecture (on which there is a recent arXiv posting with M. Scharlemann).