Function Theory of Several Complex Variables

这个项目涉及对复数和复函数的理解。自19世纪以来，复数和复变量函数已成为许多数学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。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深入理解。例如，在材料科学中，以统一的方式处理多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。结果表明，除其他因素外，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。在这个项目中进行的研究结果可能导致这些方程解的新性质的发现。这个项目有重要的教育和培训方面：一些研究生和初级数学家将积极参与这个项目。此外，首席研究员将继续组织一些复杂变量和复杂几何的国际会议，将数学家聚集在一起讨论他们的研究和教学。更确切地说，首席研究员将继续他的工作，在几个变量的复杂分析的几个基本问题，密切相关的研究微分几何，代数几何和经典动力学。更具体地说，主要研究者将继续他在几个复变量中的等价问题的研究，继续他在复空间中实子流形的全纯壳的复结构的工作，并进一步研究以CR奇点的实子流形为界的列维平坦子流形的存在性和正则性问题。他将继续研究几种复杂变量的各种刚性问题，以及它们在复杂几何和代数几何中的应用和与超刚性问题的相互作用。他将研究柯西-黎曼（CR）族可嵌入、紧致、强伪凸、三维CR流形的同时嵌入和填充问题。

This project concerns the understanding of complex numbers and complex functions. Complex numbers and functions of complex variables have become, since the nineteenth century, indispensable tools in many areas of mathematics and its application to other areas of science and engineering. The solutions of many problems in the applied sciences could ultimately depend on improvements in these complex analytic tools and a deeper understanding of their basic properties. For example, in materials science the standard method for treating multidirectional stresses in a uniform way is to represent them as complex numbers or, in more complicated situations, as complex functions. It then turns out that, among other things, 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. This project has significant educational and training aspects: several graduate students and junior mathematicians will be actively involved in this project. Also, the principal investigator will continue to organize international conferences on several complex variables and complex geometry, bringing together mathematicians to discuss their research and teaching.More precisely, the principal investigator will continue his work on several basic problems in complex analysis of several variables that are closely related to research in differential geometry, algebraic geometry and classical dynamics. More specifically, the principal investigator would like to continue his research on the equivalence problem in several complex variables, carry further his work on the complex structure of the holomorphic hull of a real submanifold in a complex space, and further his study on the existence and regularity problem for Levi-flat submanifolds bounded by real submanifolds with CR singularities. He will continue his investigation of various rigidity problems in several complex variables, as well as their applications and interactions with super-rigidity problems in complex geometry and algebraic geometry. He will investigate the simultaneous embedding and filling problem for a Cauchy-Riemann (CR) family of embeddable, compact, strongly pseudo-convex, three-dimensional CR-manifolds.