Mathematical Sciences: Differential Systems, Complex Analysis and Geometry

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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.