Mathematical Sciences: Problems in Geometric Analysis

主要研究人员将继续研究微分几何的各个领域，包括黎曼流形和Teichmuller空间的大时间热流。热流的工作将使用概率的方法来试图描述热扩散和几何量之间的关系。其中一名研究人员还将研究谱理论中一些问题的离散形式。Teichmuller理论的研究将集中在该理论在动力系统中的应用。主要研究人员还将试图发展一个Teichmuller理论的二次和多项式样的地图。该奖项将支持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空间上的解析概念之间的关系。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属性。这些数学领域在其他科学中的应用包括复杂分子的结构，液体-气体边界和宇宙的大尺度结构。

The principal investigators will continue work on various areas of differential geometry including large time heat flow on Riemannian manifolds and Teichmuller space. The work on heat flow will use method from probability to attempt to describe the relationship between heat diffusion and geometric quantities. One of the investigators will also study discrete versions of some problems in spectral theory. The research on Teichmuller theory will focus on applications of the theory to dynamical systems. The principal investigators will also attempt to develop a Teichmuller theory for quadratic and polynomial-like maps. 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.