Assistant Research Professor of Mathematics
"Quantum topology" applies ideas from mathematical physics and representation theory to the study of topological objects like knots and manifolds. The study of the Chern-Simons topological quantum field theory has been a major unifying theme, in particular its realization in terms of quantum groups.
Recently a number of researchers have worked to extend the Chern-Simons construction by passing to noncompact gauge groups. This process is still somewhat mysterious from a mathematical point of view, but seems to require using "non-semisimple" algebraic objects in place of semisimple ones. It also brings into play extra geometric structure that previously was not relevant. The resulting invariants of knots and manifolds are more complicated but also more powerful than in the compact/semisimple case. I am interested in several aspects of this problem, in particular its consequences for the study of hyperbolic knots.
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