Mathematical Sciences: Research in Several Complex Variables

首席调查员将研究几个复杂变量领域的三个项目。第一个问题是关于在高维球面上嵌入严格伪凸CR流形。在第二个项目中，他将从凸域和CR流形中推广关于小林度量的某些定理和相关概念。第三个项目涉及研究满足齐次复Monge-Ampere方程的函数，这些函数沿光滑流形具有规定的奇异行为。几十年来，曲面及其高维类似物(称为流形)如何位于多维空间中一直是数学家们非常感兴趣的课题。某些曲面，如Klein瓶，如果不强制自交，就无法放置在三维空间中。但这个表面可以被放置到四维空间中，而不需要这样的交叉点。主要研究人员将研究流形的深层次相关问题，这些流形不是基于真实的多维空间，而是基于复杂空间。

The principal investigator will study three projects in the area of several complex variables. The first concerns imbedding strictly pseudoconvex CR manifolds in higher-dimensional spheres. In the second project he will generalize certain theorems about the Kobayashi metric and related notions from convex domains and CR manifolds. The third project involves the study of functions satisfying homogeneous complex Monge-Ampere equations which have a prescribed singular behavior along a smooth manifold. How surfaces and their higher-dimensional analogues, called manifolds, may lie in multi-dimensional space has been a subject of great interest to mathematicians for many decades. Certain surfaces, like the Klein bottle, cannot be placed in three- dimensional space without forcing self intersections. But this surface can be placed into four-dimensional space without such intersections. The principal investigator will study deep related questions for manifolds which are based, not on real multi-dimensional space, but on complex spaces.