We show that closed strictly convex surfaces contracting with normal velocity equal to |A|² shrink to a point in finite time. After appropriate rescaling, they converge to spheres. We describe our algorithm to find the main test function.

We show that in each dimension n ≥ 10, there exist infinite sequences of homotopy equivalent, but mutually non-homeomorphic closed simply connected Riemannian n-manifolds with $0 \leq \rm{sec} \leq 1$, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete non-negatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of non- compact manifolds whose moduli spaces of complete metrics with sec $\geq 0$ have infinitely many connected components.

In this article, we study the group of autoequivalences of derived categories of coherent sheaves on the minimal resolution of A_n-singularities on surfaces. Our main result is to find generators of this group.

We study the local deformation space of 3-dimensional cone-manifold structures of constant curvature κ is an element of {.1, 0, 1} and cone-angles ≤ π. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem for the first L²-cohomology group of the smooth part of the cone-manifold with coefficients in the flat bundle of infinitesimal isometries. We conclude local rigidity from this. In the Euclidean case we prove that the first L²-cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms.