I study quantum and low-dimensional topology and related topics in representation theory and mathematical physics.
Most recently I been thinking about invariants of links with extra structure, for example a flat Lie algebra connection on the complement or a representation of the knot group. These are conjectured to have something to do with geometric knot invariants like hyperbolic volume.
Slightly less recently, Kyle Miller and I wrote a paper on extending some quantum-topology techniques for planar graphs to graphs embedded in higher-genus surfaces. In particular, we found a state-sum invariant of embedded graphs called the “S-polynomial.” It should be counting something (as the flow and chromatic polynomials do) but we’re not sure what. It may have to do with “defects” in lattice models.