Principal Investigator: Anton Petrunin
The principal investigator proposes to continue his research in Alexandrov geometry and its applications, studying collapse with lower curvature bound (joint with W.Tuschmann and V.Kapovitch). This part of the project can be thought of as an attempt to generalize and refine Gromov's Betti number theorem. This research can be divided into three main parts: (i) Finding new topological invariants which are finite on any family of ``similar'' manifolds which the PI has described. (ii) Using the gradient push to limit the number of bundles with the same fiber and base which can admit given lower curvature and upper diameter bounds (iii) Showing that simply connected spin manifolds with non-zero A-hat-genus can not be almost non-negatively curved. The principal investigator also proposes to continue his study of positive functions of curvature which give a bounded integral along any positively curved manifold with lower curvature bound, upper diameter and lower volume bound. This should clarify the nature of curvature tensors of Alexandrov spaces. The PI would like to show that the curvature tensor for Alexandrov spaces is well defined as a measure valued tensor in any distance co-ordinates. If true, this should give a solution to such long standing questions as whether a convex surface in Alexandrov space is an Alexandrov space.
The principal investigator proposes to compile a collection of exercises in modern geometry. The idea is to find problems which could be solved in one step. However solutions are supposed to be non trivial and would also lead to a discovery of important ideas in modern geometry. PI has already gathered some number of such problems which can be viewed online and many people had been engaged in this project. This project is oriented toward students and young scientists.
Riemannian manifold could be considered as a simplified version of space-time. The author considers an 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 resuls in the main stream direction of Riemannian geometry: how to make conclusions about global structure of space basing on local properties.
