We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the 3-dimensional case. We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.

We give a complete diffeomorphism classification of 1-connected closed manifolds M with integral homology H¤(M) ∼= Z ⊕ Z ⊕ Z, provided that dim(M) 6= 4.

We construct balanced metrics on the family of non-Khler Calabi-Yau threefolds that are obtained by smoothing after contracting ( 1, 1) -rational curves on a Khler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to the connected sum of ( 1, 1) copies of ( 1, 1) .

Let  $\sum$ be a complete minimal Lagrangian submanifold of $\mathbb{C}^n$. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of $\sum$ lies in one of these regions, then  $\sum$ is an affine space.

A Schubert class  in the Grassmannian G(k, n) is rigid if the only proper subvarieties representing that class are Schubert varieties . We prove that a Schubert class  is rigid if and only if it is defined by a partition  satisfying a simple numerical criterion. If  fails this criterion, then there is a corresponding hyperplane class in another Grassmannian G(k0, n0) such that the deformations of the hyperplane in G(k0, n0) yield non-trivial deformations of the Schubert variety . We also prove that, if a partition  contains a sub-partition defining a rigid and singular Schubert class in another Grassmannian G(k0, n0), then there does not exist a smooth subvariety of G(k, n) representing .