Mathematical Sciences: Riemannian Geometry, Geometirc Complex Analysis, and Applications

9626652格林拟议的研究涉及黎曼几何和卡勒几何。等距流形是指局部等距于黎曼流形的空间，由有限个等距群构成。为了理解非奇异黎曼流形的等距群，本文将对其进行研究。还将对满足某些几何条件的黎曼流形的拓扑类型的数目进行估计。在卡勒几何学中，研究人员将研究远距离的曲率行为如何控制空间的整体复杂结构。这是区分双曲和抛物线黎曼曲面的高维模拟。这个项目的很大一部分是关于局部几何，即每个点的小社区的几何结构，如何影响甚至控制空间的全局几何和拓扑，即空间的整体形状和性质。这对船体等大型曲面钢结构的建造具有直接的实际应用价值。例如，通过热处理将钢板变形成预先指定的曲线形状，可以用这种方法来理解。

9626652 Greene The proposed research deals with Riemannian as well as Kahler geometry. Orbifolds, spaces locally isometric to Riemannian manifolds quotiented by finite groups of isometries, are to be investigated with a view towards understanding the isometry groups of nonsingular Riemannian manifolds. Estimation of numbers of topological types for Riemannian manifolds satisfying certain geometric conditions will also be pursued. In Kahler geometry, the investigator will study how curvature behavior at large distances controls the overall complex structure of the space. This is the higher dimensional analog of distinguishing between hyperbolic and parabolic Riemann surfaces. Much of this project has to do with how the local geometry, i.e., the geometric structure in small neighborhoods of each point, influences or even controls the global geometry and topology of a space, meaning the overall shape and nature of the space. This has a direct practical application in the construction of large-scale curved steel structure such as ship hulls. The deformation of steel plates by heat treatment into pre-specified curved shapes, for example, can be understood using this approach.