Mathematical Sciences: Applications of Elliptic PDE's to Geometry and Topology

主要研究人员将研究几何尺度不变的椭圆型偏微分方程式。这样的方程自然出现在许多数学领域，包括拓扑学、几何学和数学物理。该奖项支持的三个具体研究领域是狄拉克类型算子、杨-米尔斯方程和曲率泛函。该奖项将支持在微分几何和全球分析的一般领域的研究。微分几何是研究空间几何和定义在该空间上的解析概念之间的关系的学科。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性质的学科。这些领域的数学在其他科学中的应用包括复杂分子结构、液体-气体边界和宇宙的大尺度结构。

The principal investigators will study geometric scale- invariant elliptic partial differential equations. Such equations arise naturally in many fields of mathematics including topology, geometry , and mathematical physics. The three specific areas of research supported by this award are Dirac type operators, Yang- Mills equations, and curvature functionals. 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.