Alexandrov's Geometry and Applications

AbstractAward：DMS-0406482首席研究员：Anton Petrunin首席研究员建议继续他在Alexandrov几何及其应用方面的研究，研究具有较低曲率边界的塌缩（与W.Tuschmann和V.Kapovitch联合）。这一部分的研究可以被看作是对Gromov的Betti数定理的推广和改进，主要分为三个部分：（i）寻找新的拓扑不变量，这些不变量在PI描述的任何“相似”流形上都是有限的。 (ii)利用梯度推进来限制具有相同纤维和基的丛的个数，从而使得在给定的下曲率和上直径界下，（iii）证明了具有非零A-hat-亏格的单连通自旋流形几乎不可能是非负弯曲的.本文的主要研究者还提出继续研究曲率的正函数，它给出了沿任何具有下曲率界的正弯曲流形的有界积分.直径上限和体积下限。这应该澄清thenature曲率张量的Alexandrov空间。PI想表明，曲率张量为Alexandrov空间是很好地定义为测度值张量在任何距离坐标。如果是真的，这应该给一个长期存在的问题，如是否在亚历山德罗夫空间的凸曲面是亚历山德罗夫空间的解决方案。首席研究员建议汇编一套现代几何练习。 这个想法是要找到可以一步解决的问题。 然而，解决方案被认为是不平凡的，也将导致现代几何中的重要思想的发现. PI已经收集了一些这样的问题，可以在网上查看，许多人都参与了这个项目。 这个项目面向学生和青年科学家。黎曼流形可以被认为是时空的简化版本。 本文讨论了黎曼流形的一种研究方法。它是基于考虑extremalmetrics，在适当的意义上，例如如何Riemannianmanifold崩溃到较低的dimensional对象。这种方法使我们有可能在黎曼几何的主流方向上得到新的结果：如何根据局部性质得出关于空间整体结构的结论。

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.