Facets of low-dimensional topology

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Topology is the study of objects up to rubbery stretching, and geometry the study of rigid bodies. The goal of this project is to understand certain fundamental problems in these areas by combining surprising relationships between them with deep connections to other areas of mathematics and computer science. Both topology and geometry are becoming more important to applications such as data mining, and this project includes collaboration with computer scientists as well as developing software for exploring aspects of these problems which will be freely available to other researchers via the web. This project focuses on four topics. The first is effective Mostow rigidity and torsion growth, specifically, the extent to which topological and geometric properties of hyperbolic 3-manifolds can be varied independently. The second is the computational complexity of finding the genus of a knot in the 3-sphere, in particular whether it can be determined in polynomial time. The third question is whether there exists a word-hyperbolic surface-by-surface group. The final topic is to elucidate the relationships between Heegaard-Floer homology, group orderability, and taut foliations for 3-manifolds.