GEOMETRIC STRUCTURES AND SURFACES

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The project will concentrate on the moduli of geometric structures on manifolds. This subject arose from several threads in the 19th century: (1) symmetries of crystals, which developed into the theory of crystallographic groups; (2) conformal mapping, and the corresponding holomorphic differential equations and their monodromy groups, which led to the classical theory of uniformization; (3) classical differential geometry of surfaces, generalized to Riemannian geometry, and later to Lorentzian geometry influenced by Einstein's theory of gravitation. The notion of a geometry itself was algebraicized by Felix Klein, who interpreted a geometry as the properties invariant under a transitive action of a Lie group. The present study investigates the possible ways of putting a geometric structure on a topological manifold in terms of the fundamental group and the Lie group. The goal is to describe when a given topology (often described by a combinatorial object) can be given the given geometry. In several cases, there is a complete classification of geometric structures, and the moduli space itself enjoys both a rich geometry of its own and symmetries of its own, which lead to intricate dynamical systems.The specific goals and scope of this project involve flat projective and conformal structures, especially in dimension three. The recent classification of complete affine 3-manifolds leads to many questions about more general structures. In particular the proposal asked which groups which admit proper affine actions; at present the only known hyperbolic groups are free groups. The methods involve a combination of dynamical systems, algebraic group theory, topology and geometric group theory and representation theory. Other important questions include finding obstructions for a 3-manifold to admit a projective structure; at present only one such closed 3-manifold is known. Computer experiments can help in this direction, and projects in the Experimental Geometry Lab can suggest new examples of geometric structures.