Mathematical Sciences: Groups Actions and Rigidity in Riemannian Geometry

9626232克罗克拟议的研究涉及黎曼几何领域。更具体地说，研究者计划研究基本群在非正曲率流形的泛覆盖空间边界上的诱导作用；研究负截面曲率的紧致流形的谱刚性问题；继续他对等周不等式的研究。黎曼流形是具有距离概念的空间的抽象。这些是抽象的空间，因为黎曼流形不像三维空间中的曲线和曲面那样必须位于环境空间内。这样的空间的整体形状则由其所谓的拓扑以及距离函数来确定。拟议的研究与理解拓扑和距离函数之间的这种相互作用有关。

9626232 Croke The proposed research lies in the area of Riemannian Geometry. More specifically, the investigator plans to study the induced action of fundamental groups on the boundary of the universal covering space of manifolds of non-positive curvature; to pursue spectral rigidity problems for compact manifolds of negative sectional curvature; to continue his study of isoperimetric inequalities. Riemannian manifolds are an abstraction of spaces possessing a notion of distance. These are abstract spaces in that a Riemannian manifold does not have to sit inside an ambient space, unlike curves and surfaces in 3-space. The overall shape of such a space is then determined by its so called topology as well as the distance function. The proposed research has to do with understanding this interaction between the topology and distance function.