RUI: Classical and Quantum Topology in Dimension Three

DMS-0204627

Joanna Kania-Bartoszynska

Since the introduction of quantum invariants of three dimensional

manifolds the fact that these invariants are only defined at roots of

unity has been an obstruction to analyzing their properties. However,

there is ample evidence that quantum invariants of three manifolds

exist as holomorphic functions on the unit disk that diverge

everywhere on the unit circle but at roots of unity. The investigator

uses the results of her previous research to study further quantum

invariants and to find additional applications of that research to

classical 3-manifold topology. Specifically, she works on the problem

of extending the parametric domain of the quantum 3-manifold

invariants beyond roots of unity. She also studies the application of

quantum topology to detecting symmetries of 3--manifolds and to

answering questions relating to Dehn surgery on knots.

Quantum topology is a rapidly developing area of mathematics that

brings together ideas from physics, algebra, geometry and

topology. This theory has produced a wealth of new invariants for

three dimensional manifolds. Three-manifolds are objects which locally

look like the common 3-dimensional space we live in, and 'topological

invariants' are numbers which can be associated to manifolds that

encode some information about their structure and help to classify

them. The investigator works on one of the fundamental problems in

this area, namely that of finding topological interpretations for

these new invariants.