Large Scale Geometry and Topology

大规模几何和拓扑

• 批准号: 0204576
• 负责人: Kevin Whyte
• 金额: $ 9.52万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204576&HistoricalAwards=false
• 关键词: Large; Scale; Geometry; Topology

DMS-0204576Kevin M. WhyteThis project is concerned with several aspects of large scale geometry and topology. One of the main points of study is the problem of producing actions from quasi-actions, along the lines of the classical theorems of Sullivan andTukia. Such results have significant implications in the quasi-isometric classification of groups, especially for groups with natural splittings as a complex of groups. A second area of study is large scale versions of rigidityphenomena in the geometry of symmetric spaces, and specifically the classification of quasi-isometric embeddings between symmetric spaces, giving a geometric analogue of superrigidity. A third and independant project involves coarse versions of topological rigidity, including the coarse Borel, Novikov, and Baum-Connes conjectures. Roughly speaking, large scale (or coarse) geometry is the study of geometric properties of objects "seen from far away". From this perspective, any bounded object is indistinguishable from a point, and a line of dots is indistinguishable from a solid line. This sort of geometry has been influencial recently in many areas of mathematics, notably group theory, topology, and geometric analysis. Thisresearch will explore the large scale geometry of several classes of mathematical objects, both classical geometric spaces and objects only now being viewed in a geometric manner.

kevin M. whyte该项目涉及大规模几何和拓扑的几个方面。研究的要点之一是从拟作用产生作用的问题，沿着沙利文和图基亚的经典定理的路线。这些结果在类群的准等距分类中具有重要意义，特别是对于具有自然分裂的类群作为群的复合体。第二个研究领域是对称空间几何中的大尺度刚性现象，特别是对称空间之间的准等距嵌入的分类，给出了超刚性的几何模拟。第三个独立项目涉及拓扑刚性的粗版本，包括粗Borel、Novikov和Baum-Connes猜想。粗略地说，大尺度（或粗）几何是研究“从远处看”物体的几何性质。从这个角度来看，任何有界的物体都无法与点区分，一串点与一条实线无法区分。这种类型的几何最近在数学的许多领域都有影响，特别是群论、拓扑学和几何分析。本研究将探索几类数学对象的大尺度几何，包括经典几何空间和现在仅以几何方式观察的对象。