Geometric and Analytic Problems in Several Complex Variables and Partial Differential Equations

建议：DMS-9801258主要研究人员：Linda P.Rothschild，Salah M.Baouendi摘要：主要研究复数空间中实子流形的几何以及将一个这样的流形发送到另一个流形的全纯映射。具体地说，他们将尝试对这样的映射由给定点的有限多个导数确定的流形进行分类。他们期望这项研究将导致发现这些流形的新的几何不变量、解析不变量和代数不变量。一个基本的几何和解析问题是定义在嵌入在复空间中的实超曲面的一侧上的全纯映射是否延伸到该超曲面的另一侧。更一般地，人们可以考虑复空间中高余维流形之间的光滑映射，其分量满足边界Cauchy-Riemann方程。主要研究人员将研究这种映射，特别是当流形是由多项式方程全局给出的时候，并将试图确定这种映射何时全纯地扩展到复空间。他们还将研究此类流形的代数等价和全纯等价之间的关系。自19世纪以来，复数和复变函数一直是数学中许多基本问题的重要工具，也是它在其他科学和工程领域的应用。其中一些问题的解决依赖于对这些工具及其基本属性的更好理解。例如，相对论中使用的四维时空可以被认为是一个二维复空间。对复杂空间中几何对象的变换进行定量和定性的研究，可能会对控制理论中的问题(例如，物体在特定物理约束下在空间中的运动)有新的理解和解决方案。复数分析在寻找模拟物理问题的不同方程的解方面也起着重要作用。本提案计划的研究结果可能导致发现这些方程的解的新性质，从而更好地理解相关的物理问题。

Proposal: DMS-9801258 Principal Investigators: Linda P. Rothschild, Salah M. Baouendi Abstract: The principal investigators will study the geometry of real submanifolds in complex space and the holomorphic mappings which send one such manifold into another. In particular, they will attempt to categorize those manifolds for which such mappings are determined by finitely many derivatives at a given point. They expect that this study will lead to discovery of new geometric, analytic, as well as algebraic invariants of these manifolds. A basic geometric and analytic problem is whether a holomorphic mapping defined on one side of a real hypersurface embedded in complex space extends to the other side of that hypersurface. More generally, one can consider smooth mappings between manifolds of higher codimension in complex space whose components satisfy the boundary Cauchy-Riemann equations. The principal investigators will study such mappings, in particular when the manifolds are given globally by polynomial equations, and will try to determine when such mappings extend holomorphically to the complex space. They will also study the relationship between algebraic and holomorphic equivalence of such manifolds. Complex numbers and functions of complex variables have been, since the 19th century, important tools in many fundamental problems in mathematics and its application to other areas of science and engineering. The solutions of some of these problems depend on a better understanding of these tools and their basic properties. For instance, the four dimensional space-time used in relativity theory can be considered as a two dimensional complex space. The quantitative and qualitative study of the transformations of geometric objects in complex spaces planned in this proposal may lead to new understanding and solutions of problems in control theory (e.g., motion of objects in space under certain physical constraints). Complex analysis also plays an important role in finding solutions to differenti al equations which model physical problems. Results of the research planned in this proposal could lead to the discovery of new properties of solutions of these equations and hence a better understanding of the related physical problems.