The geometry of moduli space and the mapping class group

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The goal of the project is to develop a deformation theory for closed Riemannian manifolds M of negative sectional curvature via modern invariants from symplectic geometry. Our approach is modeled on classical Teichmüller theory and the theory of Kleinian groups. We plan to replace the role of the complex structure on the two-sphere in the classical theory by the symplectic structure on the space of geodesics of the universal covering of M. This space is always symplectomorphic to the cotangent bundle of the standard sphere of dimension n-1.