My research concerns the topology and geometry of moduli spaces and applications of gauge theory to 3-manifold topology. There are two parts to this program:
(I) The generalized Casson SU(n) invariants of knots, links and 3-manifolds.
(II) The geometry and topology of moduli spaces of vector bundles over marked curves.

In part I, the goal is to derive mathematically rigorous definitions for and explicit computations of the Casson SU(n) invariants using Floer-type perturbations and SU(n) gauge theory. In part II, the goal is to study moduli spaces of parabolic bundles and parabolic Higgs bundles using methods from algebraic geometry and to apply the results to representation varieties, moduli spaces of linkages, and GIT quotients of products of Grassmanians. There is a deep connection between parts I and II (due to the analogue of the Kobayashi-Hitchin correspondence) hence the results from one part are often applicable in the other.