Alexandrov's Geometry and Applications

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AbstractAward: DMS-0406482Principal Investigator: Anton PetruninThe principal investigator proposes to continue his research inAlexandrov geometry and its applications, studying collapse withlower curvature bound (joint with W.Tuschmann andV.Kapovitch). This part of the project can be thought of as anattempt 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 thesame fiber and base which can admit given lower curvature and upper diameter bounds (iii) Showing that simply connected spin manifolds with non-zeroA-hat-genus can not be almost non-negatively curved.The principal investigator also proposes to continue his study ofpositive functions of curvature which give a bounded integralalong any positively curved manifold with lower curvature bound,upper diameter and lower volume bound. This should clarify thenature of curvature tensors of Alexandrov spaces.The PI would like to show that the curvature tensor forAlexandrov spaces is well defined as a measure valued tensor inany distance co-ordinates. If true, this should give a solutionto such long standing questions as whether a convex surface inAlexandrov space is an Alexandrov space.The principal investigator proposes to compile a collection ofexercises in modern geometry. The idea is to find problems whichcould be solved in one step. However solutions are supposed tobe non trivial and would also lead to a discovery of importantideas in modern geometry. PI has already gathered some number ofsuch problems which can be viewed online and many people had beenengaged in this project. This project is oriented towardstudents and young scientists.Riemannian manifold could be considered as a simplified versionof space-time. The author considers an approach to studingRiemannian manifolds. It is based on considering extremalmetrics, in an appropriate sense, for example how Riemannianmanifolds collapse to lower dimenssional objects. This methodmakes possible to get new resuls in the main stream direction ofRiemannian geometry: how to make conclusions about globalstructure of space basing on local properties.