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.