Function Theory of Several Complex Variables

多复变量函数论

• 批准号: 1363418
• 负责人: Xiaojun Huang
• 金额: $ 36万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1363418&HistoricalAwards=false
• 关键词: Function; Theory; Several; Complex; Variables

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.

这个项目涉及对复数和复函数的理解。自世纪以来，复数和复变函数已经成为数学许多领域中不可或缺的工具，并将其应用于其他科学和工程领域。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。结果表明，材料中裂纹的传播方向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会导致发现这些方程的解的新性质。该项目具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该项目。此外，还将继续组织多复变数和复几何的国际会议，召集数学家讨论他们的研究和教学。更准确地说，主要研究者将继续研究与微分几何、代数几何和经典动力学研究密切相关的多复变数分析中的几个基本问题。更具体地说，首席研究员想继续他的研究在多个复变量的等价问题，进一步开展他的工作的复杂结构的全纯船体的一个真实的子流形在一个复杂的空间，并进一步他的研究存在性和正则性问题的列维平坦子流形界定的真实的子流形与CR奇点。他将继续他的调查各种刚性问题在几个复杂的变量，以及它们的应用和相互作用与超刚性问题在复杂的几何和代数几何。他将研究同时嵌入和填充问题的柯西-黎曼（CR）家庭的可嵌入，紧凑，强伪凸，三维CR-流形。