Quantitative problems in Topology

University Of Chicago, Chicago IL

Investigators

Abstract

This is a proposal that aims to study the sizes of solutions to (families of) topological problems with respect to various norms. The purely metric aspect of this goes under the name of "controlled topology" and has had numerous applications to topological rigidity, stratified spaces (especially orbifolds), and homology manifolds and these will continue to be studied. However, other norms involving fewer or more derivatives are important for applications to analysis (especially variational problems) and differential geometry (e.g. systoles) --- these will be studied simultaneously with their applications. In general, topology, when applied traditionally, has been used mainly as a qualitative tool. However, it is possible to mix topological ideas with analytic ones, and fashion tools that measure not only whether things are possible or impossible, but rather how difficult they are. In so doing, topology makes contact with other branches of mathematics: notably analysis and probability. It is hoped that this will thereby enrich all these fields.