Large scale topology of metric spaces

University Of Florida, Gainesville FL

Investigators

Abstract

Large scale topology of metric spaces The project is dedicated to the large scale geometry and topology, a relatively new interdisciplinary subject which studies very large scale properties of a space. It is opposite to the classical topology which is a science of small scales, since it is based on the notion of a continuous transformation which is local by the definition. The project is devoted to the large scale counterparts of the following classic topics of topology: Dimension Theory, Cohomology Theory, Theory of Embeddings. Among the research problems selected for the project are: The quest for a large scale analog of Morita theorem that states that asymptotic dimension increases by one after crossing the space with the reals; the quest for the asymptotic analog of Alexandroff theorem connecting cohomological approach to dimension to geometric one; and a problem of quasi-isometric embedding of discrete groups into the product of trees. One of the main goals of the project is potential applications to the Novikov Higher Signature Conjecture and some other related conjectures. The classical topology has the century long history and now it is substantially developed in all directions. Its long and painful birth at the end of 19th and beginning of 20th centuries was inspired and pushed by the progress in natural sciences and mathematics, in particular in analysis. By the end of the 20th century in many areas of mathematics the necessity appeared for a discipline which would be similar to topology and which describe behavior of the large scale world. Such a discipline started to hatch in a last decade or so. This proposal is a further push for rising of such a science. There are already many connections between the asymptotic topology and applied disciplines. Thus, a part of this project is the study of the connection between large scale embeddings into the product of trees and aperiodic tilings and hence the quasi-crystals. Another connection is to the computer science. It turns out that the asymptotic infinite dimensionality of certain sequence of graphs would give the foundation for PRA ("product replacement algorithm") which is a noncomutative version of Gauss elimination algorithm. It is known that PRA works well in all instances but still it lacks a rigorous mathematical justification. Developing of the asymptotic dimension theory up to the task is the heart of the proposal.