ABSTRACT

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.

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