5850 East Hall
Phone: (734) 763-2192
Geometry and Topology
- Geometry and Topology
My research interests are mostly in differential geometry, dynamical systems and Lie groups. I am especially intrigued by problems which mix ideas and techniques from these areas. For example, one can use dynamical methods to exhibit closed geodesics on a Riemannian manifold, since they naturally correspond to the closed orbits of the manifold's geodesic flow. In general, the relationship between geometry, topology and dynamics is not well understood. Some examples of open questions are: Can the geodesic flow of a manifold of positive curvature be ergodic? Does every closed manifold support a Riemannian metric whose geodesic flow is ergodic? When are manifolds with isomorphic geodesic flows isometric? Does every closed Riemannian manifold have infinitely many closed geodesics? Incidentally, the last question was proven for the 2-sphere, using dynamical methods.
Most of my published work has focused on rigidity phenomena in geometry and dynamics. This field started with Mostows rigidity theorem: the homotopy type of a closed manifold of constant negative curvature determines its isometry type (except in dimension 2). Ideas and techniques from differential geometry, group theory and dynamical systems were all essential to the proof. Continuing fruitful interactions between these fields then led to significant advances in the study of locally symmetric spaces, manifolds of nonpositive curvature, and dynamical systems, in particular the study of actions of large groups such as SL(n,R) or SL(n,Z) for n>2. The study of homogeneous dynamical systems has been very prominent in the last two decades, and has opened up applications to other areas of mathematics, for example number theory and spectral theory.