The research is focused on the representation theory of quantum
groups at roots of 1, on invariants of 3-manifolds with flat
connections, and on some aspects of statistical mechanics. Quantum
groups emerged from the study of integrable models in quantum
field theory. These models though quite simplistic relative to
their realistic counterparts, exhibit properties which are hard to
study using conventional methods. Another class of simplistic but
extremely interesting quantum field theories are topological
quantum field theories. In these theories the "dynamics" is absent
and all they "know" in the topology of the "underlying
space-time". The theories provided new topological invariants.
"Physical" formulation involves integration over the infinite
dimensional space of connections, mathematical formulation
involves representations of quantum groups. One of the major
directions of the project is to reconcile two approaches. Another
direction of research is related to random discrete surfaces and
related structures such as random partition, random tilings of a
plane, random spanning trees on a planar graph etc. Some of the
questions in this direction are closely related to the limit when a
topological quantum field theory known as Chern-Simons theory
turns into a topological string theory.
Part of the project is focused on the
study of quantum groups at roots of unity. The goal is to
investigate the category of generic modules as a monoidal category
fibered over the corresponding group, and when it is the case, as
as a braided monoidal category fibered over a braided group. The
results will be applied to construct and study invariants of
3-manifolds with flat connections. This direction is closely
related to the possible relation between the A-polynomial and
Jones invariant of knots, which was discussed in the literature in
the last few years. The research will also be focused on dimer
models and on related models in statistical mechanics. For
example, the limit shapes for the 6-vertex model may have singular
boundaries, and one of the goals is to study the fluctuations such
singularities.
