Research Group

Geometry on Surfaces

We are interested in many kinds of geometric structures on surfaces and 3-manifolds. We want to understand the geometry of individual geometric structures and how they deform. Certain deformations give rise to dynamical systems on the moduli spaces; the ergodic theory of such systems is often very rich, offering a new perspective on the geometric objects themselves.


The dynamics of horizontal translation flow on an abelian differential can be coded by a train track (depicted in red). Both the singular flat structure and the hyperbolic metric within the same conformal class are geometric structures with intriguing moduli.

Our research projects can be summarized as:

  • Classifying horospherical orbit closures in cyclic covers of closed hyperbolic manifolds
  • Bending convex real projective structures on closed surfaces in projective 3-space
  • Ergodic theory and topological dynamics of the earthquake flow
  • Symplectic geometry of character varieties and the action of the mapping class group
  • Properties of typical affine measured laminations