The principal investigator will study the geometry of submanifolds of symmetric space through polar actions on the symmetric space. A polar action is an invariant group action on the symmetric space such that the submanifold meets the orbits of the action orthogonally. The orbits of such an action play a role in the submanifold geometry and in Morse theory on the manifold. This tool is expected to find application in the infinite dimensional case as well as the standard finite dimensional case. Symmetric spaces, such as the plane or the sphere, are among the most important spaces in geometry and physics. The principal investigator will study subspaces of symmetric spaces, for example a curve on the sphere. The major tool will be a study of the invariant motions of the symmetric space, such as the rotations of the sphere, which have a particular geometric relationship with the subspace. An understanding of this special set of motions will provide details about the geometry of the subspace itself.//
