We discuss 3-Sasakian 7-manifolds carrying metrics of positive curvature and an example of this kind on a new diffeomorphism type.

Singularities occur in Ricci flow because of curvature blowup. For dimensional reasons, when approaching a singularity, one expects the curvature to blow up like the inverse of the time to the singularity. If this does not happen, the singularity is said to be type II. The first example of a type II singularity, studied by Daskalopoulos-Del Pino-Hamilton-Sesum, occurs on a noncompact surface which is the result of capping off a hyperbolic cusp. The analysis in the surface case uses isothermal coordinates. It is not immediately clear whether it extends to higher dimensions. We look at the Ricci flow on finite-volume metrics that live on the complement of a divisor in a compact Khler manifold. We compute the blowup time in terms of cohomological data and give sufficient conditions for a type II singularity to emerge. This is joint work with Zhou Zhang.

We discuss recent progress on understanding singular special Lagrangian n-folds. Our focus will be on joint work with N. Kapouleas using gluing methods to construct a wide variety of special Lagrangian cones in every dimension three and greater.

I'll discuss joint work with Peter Petersen that shows that the Gromoll-Meyer exotic 7-sphere admits positive sectional curvature. I'll discuss the history of the problem and give a coarse outline of our solution.