Abstract: How can you put the local coordinates of a classical geometry (such as Euclidean geometry) on a space with just a topology? In the case of Euclidean geometry, this is exactly the question about which manifolds admit flat Riemannian metrics. According to Sophus Lie and Felix Klein, classical geometries are just homogeneous spaces of Lie groups. In 1938 Charles Ehresmann proposed the problem of classifying all the geometric structures modeled on a homogeneous space on a topological manifold. Forty years later this subject was rejuvenated by Bill Thurston as the context for his geometrization of 3-manifolds.
I will survey this subject, describing interesting dynamical systems arising out of the classification problem.
Please note the special time for this colloquium. A reception will follow at 2:00 in room 306.