The L^2 Geometry of the Symplectomorphism Group

In this thesis we study the geometry of the group of Symplectic diffeomorphisms of a closed Symplectic manifold M, equipped with the L^2 weak Riemannian metric. It is known that the group of Symplectic diffeomorphisms is geodesically complete with respect to this L^2 metric and admits an exponential mapping which is defined on the whole tangent space. Our primary objective is to describe the structure of the set of singularities of associated weak Riemannian exponential mapping, which are known as conjugate points. We construct examples of conjugate points on the Symplectomorphism group and solve the Jacobi equation explicitly along geodesics consisting of isometries of M. Using the functional calculus and spectral theory, we show that every such geodesic contains conjugate points, all of which have even multiplicity. A macroscopic view of conjugate points is then given by showing that the exponential mapping of the L^2 metric is a non-linear Fredholm map of index zero, from which we deduce that conjugate points constitute a set of first Baire category in the Symplectic diffeomorphism group. Finally, using the Fredholm properties of the exponential mapping, we give a new characterization of conjugate points along stationary geodesics in terms of the linearized geodesic equation and coadjoint orbits.