My primary research focus is low-dimensional manifolds and knot theory. I am especially interested in invariants of 3-manifolds and knots that are constructed using gauge theory, including Floer homology and the Casson invariant, as well as many related invariants such as knot polynomials, signature, and Khovanov homology.
Much of my recent work involves traceless character varieties of knots and links. The traceless character variety is the set of homomorphisms from the knot or link group to SU(2) which send the meridian(s) to traceless elements; often we modify the knot or link adding an "earring" with an extra condition, which insures there are no abelian homomorphisms in the character variety. There are also higher rank analogs, character varieties with a condition fixing the meridional conjugacy classes.
I also have been involved in several interdisciplinary projects related to engineering and climate science.