Alexandrov's Geometry and Applications

摘要/ abstract摘要：项目负责人：Anton petrin项目负责人建议继续他在alexandrov几何及其应用方面的研究，研究具有下曲率界的坍缩（与W.Tuschmann和v.k apovitch合作）。项目的这一部分可以被认为是尝试推广和完善格罗莫夫的贝蒂数定理。本研究可分为三个主要部分：(i)寻找新的拓扑不变量，这些拓扑不变量在PI描述的任何“相似”流形族上是有限的。(2)利用梯度推力限制了具有相同纤维和基底的束的数量，这些束可以允许给定的下曲率和上直径界限；(3)证明了非零帽属单连通自旋流形不可能几乎是非负弯曲的。在此基础上，进一步研究了曲率的正函数，给出了曲率下界、直径上界和体积下界的正曲线流形的有界积分。这将澄清亚历山德罗夫空间的曲率张量的性质。PI想要证明，亚历山德罗夫空间的曲率张量在任何距离坐标下都可以很好地定义为测量值张量。如果这是真的，那么这就解决了一些长期存在的问题，如Alexandrov空间中的凸曲面是否是Alexandrov空间。首席研究员建议编写一本现代几何习题集。这样做的目的是找到可以一步解决的问题。然而，解被认为是非平凡的，也会导致现代几何中重要思想的发现。PI已经收集了一些这样的问题，可以在网上看到，很多人都参与了这个项目。这个项目是面向学生和青年科学家的。黎曼流形可以看作是时空的简化形式。作者考虑了一种研究黎曼流形的方法。它基于在适当的意义上考虑极值度量，例如黎曼流形如何坍缩为低维对象。这种方法使黎曼几何的主流方向——如何根据局部性质得出关于空间整体结构的结论——得到新的结果成为可能。

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.