Three investigators will study problems interrelating the geometry and topology of finite dimensional manifolds with analytic, algebraic, and geometric properties of various function spaces that arise in the study of ordinary and partial differential equations on these manifolds. One investigator will study symmetry groups of a variational problem and the geometry of their associated sections and slices. The second will investigate the Grassmannian structures of the solution to integrable systems and the related Painleve analysis of these structures. And the third will analyze the relationship between the Laurent solutions to integrable systems and the geometry of their invariant tori. These three investigators will extend theories involving the geometry and algebra of certain integrable systems. One example of such systems are Hamiltonian differential equations which model the physical behavior of moving objects. These incorporate conservation of energy and momentum in an elegant manner. Solutions to these systems tend to circulate and don't have many of the properties of systems which incorporate friction.
