In the early 1980s a landmark result was obtained by Atiyah and
independently Guillemin and Sternberg: the image of the momentum map for a
torus action on a compact symplectic manifold is a convex polyhedron. Atiyah's
proof makes use of the fact that level sets of the momentum map are connected.
These proofs work in the setting of finite-dimensional compact symplectic
manifolds. One can ask how these results generalize. A well-known example of an
infinite-dimensional symplectic manifold with a finite-dimensional torus action
is 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 for
the based loop group was provided by Harada, Holm, Jeffrey and Mare in 2006.
In this thesis we study Hilbert manifolds equipped with a strong symplectic
structure and a finite-dimensional group action preserving the strong
symplectic structure. We prove connectedness of regular generic level sets of
the momentum map. We use this to prove convexity of the image of the momentum
map.Comment: 63 pages; 6 figures. Ph.D. thesis of Kathleen Smith, Department of
Mathematics, University of Toronto, 201