Geometrical Methods in Algebra and Topology

代数和拓扑中的几何方法

• 批准号: 0606002
• 负责人: Jean-Francois Lafont
• 金额: $ 7.98万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0606002&HistoricalAwards=false
• 关键词: Geometrical; Methods; Algebra; Topology

The PI proposes to work on a series of projects that incorporate techniques from algebra, geometry, and topology, with a view to applications in all three of these fields. More precisely, the PI proposes to work on the following four circles of ideas:(1) Answering various classical questions in topology (such as the Novikov conjecture, the Borel conjecture, computation of the algebraic K-theory) under suitable geometric hypotheses on the spaces.(2) Working on problems related to the quasi-isometric classification of finitely generated groups.(3) Exploring the boundary between the space of CAT(-1) metrics and the space of negatively curved Riemannian metrics on a fixed manifold.(4) Studying the topology (cohomology, bounded cohomology, submanifolds) of locally symmetric spaces.These projects are intended to exhibit the interdependence of methods and results between the three fields of algebra (primarily group theory), geometry (negatively curved spaces) and topology (both geometric and algebraic).Groups are sets equipped with a "multiplication", and classically arose as symmetries of underlying geometric objects. Geometry is the study of properties of spaces equipped with some additional structure, usually of a metric nature (i.e. allowing one to measure distances in the space). In the field of topology, one forgets about the underlying distance, and only retain the notion of points in the space being "close together". While these three fields have always had some overlap, research from the last twenty years have led to an increasingly fertile exchange of ideas and applications amongst them. The PI's research is intended to further develop the links between these three fields.

PI建议开展一系列项目，将代数，几何和拓扑学的技术结合起来，以期在所有这三个领域中应用。 更确切地说，PI提出了以下四个思想圈：（1）在空间上适当的几何假设下，解决拓扑学中的各种经典问题（如诺维科夫猜想，波莱尔猜想，代数K理论的计算）。(2)研究与群的拟等距分类有关的问题。(3)探讨了固定流形上CAT（-1）度量空间与负曲黎曼度量空间的边界。(4)研究局部对称空间的拓扑（上同调，有界上同调，子流形）。这些项目旨在展示代数（主要是群论），几何（负弯曲空间）和拓扑（几何和代数）三个领域之间的方法和结果的相互依赖性。群是配备了“乘法”的集合，经典地作为基本几何对象的对称性而出现。 几何学是研究具有某种额外结构的空间的性质，通常具有度量性质（即允许人们测量空间中的距离）。 在拓扑学领域，人们忘记了潜在的距离，只保留了空间中的点“靠近”的概念。 虽然这三个领域总是有一些重叠，但过去二十年的研究已经导致它们之间的思想和应用的交流越来越丰富。 PI的研究旨在进一步发展这三个领域之间的联系。