Abstract Proposal: DMS-9803051 Principal Investigators: Eugene Lerman and Susan Tolman The issues being investigated in this study of Hamiltonian systems include: classifying symplectic manifolds with Hamiltonian group action, determining when such manifolds admit Kaehler structures, computing their cohomology rings, the capacities of symplectic quotients, intersection cohomology of singular quotients, and the two-body problem on the sphere. The special case of symplectic manifolds with large torus actions is particularly amenable to study. Working with Y. Karshon, S. Tolman has reduced the classification question to combinatorics and homotopy information in the case where the torus is of dimension which is one less than half the dimension of the manifold. The principal investigators plan to use these methods to find necessary and sufficient conditions for such a space to admit an invariant Kaehler structure. S. Tolman also hopes to use these techniques and recent results in symplectic topology to extend the results to tori of dimension two less than half the dimension of the manifold. The investigators also plan to address the question of invariants of reduced spaces, and together with T. Tokieda they will investigate the capacity of symplectic quotients. They will also study the cohomology of reduced spaces, both at regular values (joint work by S. Tolman and J. Weitsman), and at singular values (joint with J. Ho.) Finally, E. Lerman (jointly with T. Tokieda) will study the two body problem on the two sphere -- a limiting case of two coupled rigid bodies. The techniques include classifying relative equilibria, the energy-momentum method, and numerical simulation. On the scale of everyday life, essentially all objects follow paths determined by the laws of classical mechanics. In many cases -- for example, a spinning top, two coupled rigid bodies, a satellite with flexible attachments, or the solar system itself -- the mechanical system possesses a great deal of symmetry. In these cases, for each degree of symmetry there is a conserved quantity, called ``momentum''. Linear and angular momentum are the most common examples of this phenomena. A great deal can be learned about such systems both by studying the momentum, and by ``dividing out'' by the set of symmetries to study a simpler system. We study systems which, while not necessarily examples of concrete physical systems, abstract out many of the common features of such systems. We intend to study, among other questions, what happens when systems have very large amounts of symmetry, how to deal with singularities which sometimes arise in ``dividing out'' by symmetries, and how to use the momentum to answer important questions about the system. Our work,including both the results and some of the methods we plan to develop, should be useful to scientists and engineers studying concrete physical systems.
