Applications of quantum groups to 4-dimensional topological field theory

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Recently, Ben-Zvi, Brochier, and Jordan [BZBJ] have introduced a 4-dimensional topological field theory - dubbed Betti Geometric Langlands (BGL) theory - intimately related to N=4 super Yang-Mills theory, and to the moduli spaces of Betti local systems on surfaces and 3-manifolds. The BGL theory produces among other things, quantizations of the Poisson structures on spaces of $G$-local systems of Riemann surfaces.It is an interesting an important problem to quantize individually the symplectic leaves of these moduli spaces, which correspond to local systems whose monodromy around each puncture lies in a prescribed conjugacy class, and to leverage the output to produce new topological invariants. Specific projects include:1) Constructing quantizations of conjugacy classes in reductive groups, as co-ideal sub-algebras, and developing the resulting braided module categories for the quantum group2) Studying topological field theories on orbifolds, and relating these to quantum symmetric pairs.3) Studying special features of these and related constructions when the quantum parameter is a root of unity.