I will discuss recent work with Akutagawa and Carron concerning the solvability of the Yamabe problem to compact stratified spaces. There are many new obstructions to solvability, some of which we identify quite explicitly. We also discuss regularity of the solutions along the singular strata, and identify some possible rigidity phenomena for cases where existence fails.

I will give a brief introduction to mean curvature flow (MCF) of hypersurfaces and survey recent progress with Toby Colding on the dynamics of mean curvature flow near a singularity. MCF is a nonlinear heat equation where the hypersurface evolves to minimize its surface area and the major problem is to understand the possible singularities of the flow and the behavior of the flow near a singularity.

We shall indicate the rough idea that a hypersurface transversal to a Killing vector field that flows by MCF remains transverse to it and will study this evolution in some warped products of the real line times a riemannian manifold. This talk contains joint work with A. Borisenko.

The aim of this talk is to present some new form of the weak (and Omori-Yau) maximum principle for general linear operators on a Riemannian manifold, notably trace operators, that make particularly clear the function theoretical aspects of this type of results and free them from the request (at least for the weak case) of completeness fo the underlying metric. We also discuss applications to gradient Ricci solitons and to the geometry of immersions to show the versatility and powerfulness of the tools when applied to geometric problems.

We review some stability and area minimizing properties for minimal and constant mean curvature surfaces in the euclidean 3-space. We will present the case of complete surfaces, free boundary and the prescribed symmetries one. In particular, we will prove that area minimizing surfaces in some quotients of $\mathbb{R}^3$ are planar.

Hamilton's compactness theorem is one of the most fundamental tools in the study of Ricci flow. Extensions of this result are often required. However, the precise extension which has been used in the most famous applications of Ricci flow is not quite right as we demonstrate by giving a counterexample. The main new idea required here also leads to quite different new applications in the study of unbounded curvature Ricci flows.