When can a global topology support a local structure modeled on a classical geometry? A "classical geometry" means the structure of a manifold invariant under a transitive action of a Lie group. For a fixed topology, the space of such structures is a natural object, with a rich geometry and symmetry of its own, associated with the topology and the homogeneous space. The study of deformation spaces of geometric structures was initiated by Charles Ehresmann in the 1930's. It unifies what have been disparate areas of research in the 19th century (crystallography, holomorphic differential equations and conformal mapping, development of projective geometry and non-Euclidean geometry). The subject became prominent through the influence of William Thurston in the 1970's. The prototype of this theory is the space of hyperbolic geometry structures on a closed surface, which by the classical uniformization theorem, identifies with the Teichmueller space of the surface. Our project explores three recent developments - geometric structures on 3-manifolds, higher Thurston-Teichmueller theory, and Anosov representations - where the rich structure of Teichmueller space generalizes to deformation spaces of more complicated geometries. The intimate relations of this subject with many other fields of mathematics underscores its central role in mathematics.
This project synthesizes disparate mathematical subjects: the topology of manifolds, various kinds of geometry, algebra and dynamics. The Moebius band is an example of a two dimensional manifold with only one side. It describes, for example, the collection of all straight lines in the plane. The universe we live in is an example of a three dimensional manifold. The position and velocity of a satellite or missile is described by a point in a six-dimensional manifold. Different kinds of geometries distinguish special properties of manifolds. The Moebius band is naturally described using the projective geometry inspired by the work of Renaissance painters. Cartographers used conformal geometry to produce more accurate maps of the world. Differential geometry enabled Einstein to develop his theory of gravitation. Chemists use the algebra of groups to classify crystals. The periodic table of chemical elements is intimately connected to the group of rotations of space. Much of this mathematical landscape remains unexplored. Using modern computers, students can contribute to this investigation. The exploration of explicit examples and their interactions provides problems for talented students, inviting them to the excitement of mathematical research.
