Mathematical Sciences: Some Natural Problems in Complex Geometry

邱教授将研究复杂几何中的几个基本问题，包括复杂平台问题的解决方案和三维CR流形的嵌入问题。他还将尝试将以前关于有限维李代数和孤立超曲面奇点的工作推广到更高的余维奇点。该奖项将支持复杂微分几何的三个基本领域的研究。复杂的高原问题是一个几何极小化问题。本研究的目的是对作为复流形边界的奇维实数子流形进行分类。这项工作与CR流形的研究密切相关--CR流形空间通常作为复空间的子空间出现。虽然这项建议所支持的大部分工作都属於所谓的“纯数学”范畴，但邱教授在这方面的大部分工作已在控制理论和滤波方面得到应用。因此，这项工作提醒人们基础研究和应用之间的联系。

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.