Mathematical Sciences: Some Natural Problems in Complex Geometry

丘教授将研究复杂几何中的几个基本问题，包括复杂高原问题的解法和三维CR流形的嵌入问题。他还将尝试将之前关于有限维李代数和孤立超曲面奇点的研究扩展到更高的余维奇点。该奖项将支持复杂微分几何三个基本领域的研究。复杂平台问题是一个几何极小化问题。本文的研究目标是对作为复流形边界的奇维实子流形进行分类。这一工作与通常作为复空间子空间出现的CR流形空间的研究密切相关。虽然本提案支持的大部分工作属于所谓的“纯数学”领域，但丘教授在该领域的许多工作已在控制理论和滤波中得到应用。因此，这项工作提醒人们注意基础研究与应用之间的联系。

Professor Yau will study several fundamental problems in complex geometry including solutions to the complex Plateau problem and the embedding problem for three dimensional CR manifolds. He will also try to extend previous work on finite dimensional Lie algebras and isolated hypersurface singularities to higher codimension singularities. This award will support research in three basic areas of complex differential geometry. The complex Plateau problem is a geometric minimization problem. The goal of the proposed research is to classify those real odd-dimensional real submanifolds which are boundaries of complex manifolds. This work is closely connected to research on CR manifolds - spaces which usually occur as subspaces of complex space. Although most of the work which will be supported by this proposal falls into the area of what has been called "pure mathematics", much of Professor Yau's work in this field has found application in control theory and filtering. The work thus serves as a reminder of the link between basic research and application.