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

多复变量和偏微分方程的几何和解析问题

• 批准号: 9801258
• 负责人: Linda Rothschild
• 金额: $ 28.95万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801258&HistoricalAwards=false
• 关键词: Geometric; Analytic; Problems; Several; Complex

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.

提案：DMS-9801258 首席研究员：Linda P. Rothschild、Salah M. Bauuendi 摘要：首席研究员将研究复杂空间中真实子流形的几何形状以及将一个流形发送到另一个流形的全纯映射。特别是，他们将尝试对那些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几何、解析以及代数不变量。一个基本的几何和分析问题是在嵌入复杂空间的真实超曲面的一侧定义的全纯映射是否延伸到该超曲面的另一侧。更一般地，我们可以考虑复杂空间中较高余维流形之间的平滑映射，其分量满足边界柯西-黎曼方程。主要研究人员将研究此类映射，特别是当流形由多项式方程全局给出时，并将尝试确定此类映射何时全纯扩展到复空间。他们还将研究此类流形的代数和全纯等价之间的关系。 自 19 世纪以来，复数和复变量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的应用的重要工具。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，相对论中使用的四维时空可以被视为二维复空间。该提案计划对复杂空间中几何对象的变换进行定量和定性研究，可能会带来对控制理论问题（例如，在某些物理约束下空间中对象的运动）的新理解和解决方案。复分析在寻找模拟物理问题的微分方程的解方面也发挥着重要作用。该提案中计划的研究结果可能会发现这些方程解的新性质，从而更好地理解相关的物理问题。