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

提案：DMS-9801258主要研究者：琳达P. Rothschild、Salah M.巴文迪 摘要：主要研究人员将研究几何的真实的子流形在复杂的空间和全纯映射发送到另一个这样的流形。特别是，他们将试图对那些映射由给定点上的许多导数确定的流形进行分类。他们希望这项研究将导致发现新的几何，分析，以及代数不变量的这些流形。一个基本的几何和分析问题是定义在复空间中的真实的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，可以考虑复空间中的高余维流形之间的光滑映射，其分量满足边界柯西-黎曼方程。主要研究人员将研究这样的映射，特别是当流形是由多项式方程全局给定时，并将试图确定何时这样的映射全纯扩展到复空间。他们还将研究这种流形的代数等价和全纯等价之间的关系。 自世纪以来，复数和复变函数一直是数学中许多基本问题及其在科学和工程其他领域应用的重要工具。其中一些问题的解决方案取决于对这些工具及其基本属性的更好理解。例如，在相对论中使用的四维时空可以被认为是一个二维的复空间。该提案中计划的复杂空间中几何对象变换的定量和定性研究可能会导致对控制理论中问题的新理解和解决方案（例如，在某些物理约束下的空间物体运动）。复分析在求解模拟物理问题的微分方程中也起着重要的作用。本提案中计划的研究结果可能会导致发现这些方程的解的新特性，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.