Large Scale Geometry and Topology

DMS-0204576凯文M. WhyteThis项目涉及大规模几何和拓扑的几个方面。 研究的要点之一是问题的生产行动准行动，沿着线的经典定理沙利文andTukia。 这样的结果有显着的意义，在拟等距分类的群体，特别是与自然分裂的群体作为一个复杂的群体。 第二个研究领域是对称空间几何中刚性现象的大规模版本，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。 第三个和独立的项目涉及粗糙版本的拓扑刚性，包括粗糙的博雷尔，诺维科夫和鲍姆-康纳斯拓扑。 粗略地说，大尺度（或粗糙）几何是研究“从远处看”的物体的几何特性。 从这个角度来看，任何有界对象都无法与点区分开来，而一条点线与实线也无法区分开来。 这种几何最近在数学的许多领域都很有影响力，特别是群论、拓扑学和几何分析。 本研究将探讨几类数学对象的大尺度几何，包括经典的几何空间和现在才以几何方式观察的对象。

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.