The principal investigator will use fiber bundle theory systematically to study Hamiltonian mechanics and symplectic geometry, bifurcation of relative equilibria, linearization of flows along orbits, optimal control of deformable bodies, and the geometry of the group of symplectic diffeomorphisms and of Teichmuller space. Modern symplectic geometry has been used to analyze problems arising from the study of mechanical systems. The principal investigator will use one of the new tools from this geometry known as "fiber bundle theory." In use for the problems at hand, these are bundles of symmetric groups or movements which are mathematically attached to each point of a surface or hypersurface. An elementary example would be a reflection of each plane tangent to a given sphere.
