Large Scale Geometry and Topology

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

DMS-0204576 Kevin M. Whyte This 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 and Tukia. 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 rigidity phenomena 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. This research 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.