Function Theory of Several Complex Variables

翻译后摘要：主要研究者建议在这个项目中的一些问题，在该地区的几个复变量和柯西-黎曼分析。这些问题不仅从复分析的角度来看是重要的，而且从非线性分析，偏微分方程，复奇异性理论和机械工程的角度来看也是重要的。更具体地说，PI希望进一步了解复杂的分析和复杂的奇异性理论中的各种刚性现象。PI想继续他以前的研究工作，研究嵌入在复欧氏空间中的真实的子流形的全纯不变理论。研究复流形中真实的子流形的全纯理论与经典力学、非线性偏微分方程和经典动力学中许多著名问题的内在联系。PI的目的是实现同时嵌入问题的CR族的可嵌入紧强伪凸三维CR流形。从世纪以来，复数和复变函数理论已经成为数学的许多领域及其在其他科学和工程领域的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以及对其基本性质的更好理解。例如，在材料科学中，以统一的方式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，表示为复函数。例如，材料中裂纹的传播方向与与这些复数或函数相关的某些方程的性质有关。在该项目中进行的研究结果可能会导致发现这些方程的解的新特性，然后转化为对材料相关机械特性的更深入理解。

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.