Mathematical Sciences: Topics in Riemannian and Complex Geometry

主要研究者将研究全局黎曼几何中与黎曼流形的存在、分类和退化有关的几个问题，这些流形对于曲率的积分范数是极值的。他们还将研究这一理论在Kahler几何、微分拓扑和数学物理中的应用。一位首席研究员将研究临界度量问题退化解的存在性。第二个主要研究者将研究正标量曲率的完全流形的完整性。该奖项将支持微分几何和全局分析领域的研究。微分几何是研究空间几何与空间上定义的解析概念之间关系的学科。全局分析是通过拼凑局部信息来研究空间的整体几何和拓扑特性。这些数学领域在其他科学中的应用包括复杂分子的结构、液气边界和宇宙的大尺度结构。

The principal investigators will study several problems in global Riemannian geometry related to the existence, classification, and degeneration of compact Riemannian manifolds which are extremal for an integral norm of the curvature. They will also study applications of this theory to Kahler geometry, differential topology, and mathematical physics. One principal investigator will study the existence of degenerate solutions to the critical metric problem. The second principal investigator will study the holonomy of complete manifolds of positive scalar curvature. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.