Mathematical Sciences: Differential Systems, Complex Analysis and Geometry

这个项目涉及几个复杂变量、外部微分系统和微分几何中的许多主题。特别是，研究者计划研究Cauchy-Riemann嵌入问题，曲率处方问题，对称空间上的Radon变换及其在等谱刚性中的应用，弱伪凸域上的核和全纯扩展问题。微分几何和复分析中的许多问题都可以用外部微分系统的语言来表达——外部微分系统是偏微分方程系统的一种推广，允许它们在弯曲空间上全局定义，而不必引入局部坐标。提出的几个问题涉及外部系统理论的应用。特别是曲率处方问题，通过在光滑但其他方面是裸露的空间上求解相关的外部方程组来寻求满足给定曲率属性集的空间。

9625633 Kuranishi This project deals with a number of topics in several complex variables, exterior differential systems and differential geometry. In particular, the investigator plans to pursue the Cauchy-Riemann embedding problem, curvature prescription problems, the Radon transform on symmetric spaces and its application to isopectral rigidity, and kernels on weakly pseudoconvex domains and holomorphc extension problems. Many problems in differential geometry and complex analysis can be couched in the language of exterior differential systems - exterior differential systems are a generalization of systems of partial differential equations allowing them to be defined globally on curved spaces without having to introduce local coordinates. Several of the proposed problems deal with applications of exterior systems theory. The curvature prescription problem, in particular, seeks to find spaces satisfying a given set of curvature properties by solving an associated system of exterior equations on a smooth but otherwise bare space.