Mathematical Sciences: Submanifold Geometry and Polar Actions

主要的研究者将通过在对称空间上的极作用来研究对称空间的子流形的几何。极作用量是对称空间上的不变群作用量，以子流形与作用量的轨道垂直相交。这种作用量的轨道在子流形几何和流形上的莫尔斯理论中起着重要作用。该工具可望在无限维情形和标准有限维情形中得到应用。对称空间，如平面或球面，是几何学和物理学中最重要的空间之一。首席研究员将研究对称空间的子空间，例如球面上的曲线。主要的工具将是研究对称空间的不变运动，例如球体的旋转，它与子空间有特殊的几何关系。对这组特殊运动的理解将提供有关子空间本身几何结构的细节。//

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.//