My research interests include low-dimensional topology, knot theory, symplectic geometry, and gauge theory, as well as various interdisciplinary areas.
Several current projects are related to traceless SU(2) character varieties of knots and links, as well as higher rank analogs. There are a variety of interesting ways to decompose knot complements along a surface, splitting it into tangles. The traceless character varieties of the tangles produce Lagrangian submanifolds in the traceless character variety of the splitting surface, which is symplectic. The Fukaya category of this symplectic space provides a very useful algebraic framework to study these situations. These character varieties are related to singular instanton knot Floer homology, Khovanov homology and Casson-Lin invariants.
For more detail about my research and publications.