The project contains the following main topics: 1. Approximation of Riemannian manifolds by polyhedral metrics and generalization of Alexandrov embedding theorem. The PI has proved that approximability by polyhedral metrics implies a new kind of curvature bound which appears naturally but has never been considered in geometry before. The proof is interesting in its own right, as it uses a Lemma closely resembling the Alexandrov embedding theorem (which states that any positively curved metric on a 2-sphere is isometric to a convex surface in a Euclidean space). This brings us back to a more geometrical point of view on Riemannian Geometry, according to which ``interesting'' curvature bounds should arise from properties of embeddings of manifolds into Euclidean space. This circle of ideas also gives a new approach to the old conjecture that every simply connected Riemannian manifold with positive curvature operator is diffeomorphic to a sphere; 2. Collapsing with lower curvature bound. The general goal of this topic is to understand how collapsing with lower curvature and diameter bounds happens and, in the very best case, to construct a structure analogous to the one obtained by Cheeger-Fukaya-Gromov for the case of bounded curvature. 3. Gate spaces. Gate spaces are related to a circle of problems arising from the question of K.Grove of whether there is an Alexandrov space that has two different smoothings into Riemannian manifolds of the same dimension and lower curvature bound; 4. Applications of megafolds to collapsing with bounded curvature and to Ricci flow. Megafolds are a generalization of Riemannian manifolds and orbifolds that has already proved it usefulness for collapsing with bounded curvature; in particular, they were used by the PI to prove, in coloboration with W.Tuschmann, the main part of the Klingenberg-Sakai Conjecture. Such a collapsing also arises naturally from the rescaling of the Ricci flow; it can be used to construct singularity models for the Ricci flow with no injectivity radius estimates; 5. Theory of Alexandrov spaces. Alexandrov spaces appear naturally as limits of Riemannian manifolds with lower curvature bound. Most geometric results which are true for Riemaninan manifolds with lower curvature bound are also true for Alexandrov spaces; however, there are several such results that cannot be generalized. For example, it is not known whether a convex hypersurface in an Alexandrov space is also an Alexandrov space. Such problems are mostly due to the lack of local analysis, and that is what the PI proposes to study.
Riemannian manifold, which could be considered as a simplified version of space-time, is a way too complicated object. The first topic in this proposal is aimed at studing Riemanian manifolds by means of approximation by simpler objects. These objects are polyhedral spaces, i.e. spaces glued of Euclidean polyhedra. The other topics considers a different approach to studing Riemannian manifolds. It is based on considering extremal metrics, in an appropriate sense, for example how Riemannian manifolds collapse to lower dimenssional objects. This method makes possible to get new results in the main stream direction of Riemannian geometry: how to make conclusions about global structure of space basing on local properties.
