Function Theory of Several Complex Variables

这个项目涉及对复数和复数函数的理解。自十九世纪以来，复变量的复数和复变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，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。结果表明，材料中裂纹扩展的方向与某些方程的性质有关，这些方程与这些复数或函数有关。本课题研究的结果可能会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：几个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关于几个复变量和复杂几何的国际会议，聚集数学家讨论他们的研究和教学。更准确地说，首席研究员将继续他的工作，在几个变量的复变分析中的几个基本问题上，这些问题与微分几何、代数几何和经典动力学的研究密切相关。更具体地说，主要研究人员希望继续研究多个复变量的等价问题，进一步研究复空间中实子流形的全纯壳的复结构，并进一步研究由具有CR奇点的实子流形限定的Levi平坦子流形的存在性和正则性问题。他将继续研究多个复变量中的各种刚性问题，以及它们在复几何和代数几何中的应用及其与超刚性问题的相互作用。他将研究Cauchy-Riemann(CR)族的同时嵌入和填充问题，该族是可嵌入的、紧的、强伪凸的三维CR-流形。

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.