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In the early $1980$s a landmark result was obtained by Atiyah andindependently Guillemin and Sternberg: the image of the momentum map for atorus action on a compact symplectic manifold is a convex polyhedron. Atiyah'sproof makes use of the fact that level sets of the momentum map are connected.These proofs work in the setting of finite-dimensional compact symplecticmanifolds. One can ask how these results generalize. A well-known example of aninfinite-dimensional symplectic manifold with a finite-dimensional torus actionis the based loop group. Atiyah and Pressley proved convexity for this example,but not connectedness of level sets. A proof of connectedness of level sets forthe based loop group was provided by Harada, Holm, Jeffrey and Mare in $2006$.  In this thesis we study Hilbert manifolds equipped with a strong symplecticstructure and a finite-dimensional group action preserving the strongsymplectic structure. We prove connectedness of regular generic level sets ofthe momentum map. We use this to prove convexity of the image of the momentummap.

Connectivity and Convexity Properties of the Momentum Map for Group Actions on Hilbert Manifolds