Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

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The research is focused on the representation theory of quantumgroups at roots of 1, on invariants of 3-manifolds with flatconnections, and on some aspects of statistical mechanics. Quantumgroups emerged from the study of integrable models in quantumfield theory. These models though quite simplistic relative totheir realistic counterparts, exhibit properties which are hard tostudy using conventional methods. Another class of simplistic butextremely interesting quantum field theories are topologicalquantum field theories. In these theories the "dynamics" is absentand all they "know" in the topology of the "underlyingspace-time". The theories provided new topological invariants."Physical" formulation involves integration over the infinitedimensional space of connections, mathematical formulationinvolves representations of quantum groups. One of the majordirections of the project is to reconcile two approaches. Anotherdirection of research is related to random discrete surfaces andrelated structures such as random partition, random tilings of aplane, random spanning trees on a planar graph etc. Some of thequestions in this direction are closely related to the limit when atopological quantum field theory known as Chern-Simons theoryturns into a topological string theory.Part of the project is focused on thestudy of quantum groups at roots of unity. The goal is toinvestigate the category of generic modules as a monoidal categoryfibered over the corresponding group, and when it is the case, asas a braided monoidal category fibered over a braided group. Theresults will be applied to construct and study invariants of3-manifolds with flat connections. This direction is closelyrelated to the possible relation between the A-polynomial andJones invariant of knots, which was discussed in the literature inthe last few years. The research will also be focused on dimermodels and on related models in statistical mechanics. Forexample, the limit shapes for the 6-vertex model may have singularboundaries, and one of the goals is to study the fluctuations suchsingularities.