The Smale Conjecture (1961) may be stated in any of the following equivalent forms:

        • The space of embedded 2-spheres in R^3 is contractible.

        • The inclusion of the orthogonal group O(4) into the group of diffeomorphisms of the 3-sphere is a homotopy equivalence.

        • The space of all Riemannian metrics on S^3 with constant sectional curvature is contractible.

While the analogous statement one dimension lower can be proven in many ways --- for instance using the Riemann mapping theorem --- Smale's conjecture turned out to be surprisingly difficult, and remained open until 1983, when it was proven by Hatcher using a deep combinatorial argument. Smale's Conjecture has a natural generalization to other 3-dimensional space forms: if M is a Riemannian 3-manifold of nonzero constant sectional curvature, then the inclusion of the isometry group into the diffeomorphism group is a homotopy equivalence. The lecture will explain how Ricci flow through singularites, as developed in the last few years by John Lott, Richard Bamler, and myself, can be used to address this conjecture.

This is joint work with Richard Bamler.