Function Theory of Several Complex Variables

多复变量函数论

• 批准号: 0500626
• 负责人: Xiaojun Huang
• 金额: --
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500626&HistoricalAwards=false
• 关键词: Function; Theory; Several; Complex; Variables

Abstract:The principal Investigator proposes in this project to work on a number of problems in the area of several complex variables and Cauchy-Riemann Analysis. These problems are not just important from the point of view of complex analysis, but also from the perspective of non-linear analysis, partial differential equations, complex singularity theory and mechanical engineering. More specifically, the PI wishes to further the present understanding of various rigidity phenomena in complex analysis and complex singularity theory. The PI would like to continue his previous investigation on the holomorphic invariant theory for real submanifolds embedded in the complex Euclidean spaces. The investigation of the intrinsic connections of the holomorphic theory of real submanifolds in a complex manifold with many well-known problems in classical mechanics, non-linear partial differential equations and classical dynamics will also be pursued. The PI intends to carry out the simultaneous embedding problem for a CR family of embeddable compact strongly pseudoconvex three-dimensional CR manifolds. Various geometric and analytic properties for Cauchy-Riemann mappings will be studied, too.Complex numbers and the theory for functions with complex variables have become, since the 19th century, indispensable tools in many areas of mathematics and in its application to other areas of science and engineering. Indeed, the solutions of many problems in the appliedsciences could ultimately depend on improvements in these complex analytic tools and a better understanding of their basic properties. For instance, in material science, the standard method for treating multi-directional stresses in a uniform way is to represent them as complex numbers or, in more complicated situations, as complex functions. It then turns out, for instance, that the direction of the propagation of cracks in materials is related to the properties of certain equations associated with these complex numbers or functions. Results of the research to be carried out in this project may lead to the discovery of new properties of solutions of these equations, which would then translate to a deeper understanding of the related mechanical properties of materials.

摘要：主要研究人员在这个项目中提出了在多复变和柯西-黎曼分析领域的一些问题的工作。这些问题不仅从复分析的角度具有重要意义，而且从非线性分析、偏微分方程组、复奇点理论和机械工程的角度也具有重要意义。更具体地说，PI希望加深目前对复分析和复奇点理论中各种刚性现象的理解。PI愿意继续他之前对嵌入在复欧氏空间中的实子流形的全纯不变理论的研究。研究复流形中实子流形的全纯理论与经典力学、非线性偏微分方程组和经典动力学中的许多著名问题之间的内在联系。PI研究了一族可嵌入的紧致强伪凸三维CR流形的同时嵌入问题。课程还将研究柯西-黎曼映射的各种几何和解析性质。自19世纪以来，复数和复变量函数理论已成为许多数学领域及其在其他科学和工程领域应用中不可或缺的工具。事实上，应用科学中许多问题的解决最终可能取决于这些复杂分析工具的改进和对其基本性质的更好理解。例如，在材料科学中，统一处理多方向应力的标准方法是将它们表示为复数，或者在更复杂的情况下表示为复杂函数。例如，结果表明，材料中裂纹扩展的方向与与这些复数或函数相关的某些方程的性质有关。本项目将开展的研究结果可能导致这些方程解的新性质的发现，这将转化为对材料相关力学性质的更深层次的理解。