|Three and Four Dimensional Topology and Geometry.
My work in this area (mostly joint with Paul Kirk of Indiana University) has involved investigating properties of three and four dimensional manifolds, including knots in the three-sphere, using the techniques of gauge theory and by studying representations of their fundamental groups in certain Lie groups. In particular we have developed techniques of computing spectral flow and Chern-Simons invariants using representation spaces and algebraic topology.
Riemann Surfaces and Their Moduli Spaces.
My interests involving Riemann surfaces include: moduli spaces, Hurwitz spaces, Grothendieck's theory of Dessins d'Enfants, and the uniformization of Riemann surfaces.
Computerized Shape Recognition
In a collaboration with Anuj Srivastava and Gordon Erlebacher, I am working on some geometry problems related to computerized shape recognition (with face recognition as the ultimate goal).