Coarse Topology of Metric Spaces

University Of Florida, Gainesville FL

Investigators

Abstract

DMS-0305152 Alexander Dranishnikov The project is dedicated to the asymptotic (coarse) topology. There is a macro-micro analogy which helps to explore and navigate the relatively new world of coarse topology. The P.I. is planning to work in the asymptotic counterparts of the following classic topics of topology: Dimension Theory, Cohomology Theory, Theory of Embeddings. One of the main goals of his research is potential applications to the Novikov Higher Signature Conjecture and some other related conjectures. Asymptotic topology is a relatively new subject which studies the large scale properties of a space. The classical topology is a science of a very small scale, since it is based on the notion of a continuous transformation which is local. 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 science.