Function Theory of Several Complex Variables

多复变量函数论

• 批准号: 2247151
• 负责人: Xiaojun Huang
• 金额: $ 36.44万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247151&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 are indispensable tools in many areas of mathematics and have deep applications to other areas of science and engineering. The solutions of many problems in applied sciences could ultimately depend on improvements in complex analytic tools. Results of the research to be carried out in this project may lead to the discovery of novel properties of complex-valued functions. This project also has significant educational and training aspects. Graduate students, undergraduate students, and junior researchers will be actively involved in the project. Also, the principal investigator will continue to organize international conferences on several complex variables and complex geometry, bringing together many mathematicians to discuss their research and teaching. This project involves work on a number of problems in the broad area of several complex variables and Cauchy-Riemann geometry. The problems under consideration also have connections to other mathematical fields, including differential geometry, complex singularity theory, algebraic geometry, and classical dynamics. More specifically, the PI will continue his investigation of rigidity problems in several complex variables, along with their applications and interactions with complex geometry and algebraic geometry. He will continue his research on the equivalence problem in several complex variables, pursue his ongoing study of the complex structure of the holomorphic hull of a real submanifold in a complex space, and further his investigations on the existence and regularity problem for Levi-flat submanifolds bounded by real submanifolds with CR singularities. In addition, he will continue his work on boundary invariants of weakly pseudo-convex domains, finite type conditions and the Bloom conjecture, and canonical metrics on Stein spaces with isolated normal singularities. Finally, he will continue his study on the boundary unique continuation problems for holomorphic functions and his work on transversality problems in the Cauchy-Riemann setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目涉及对复数和复函数的理解。复数和复变量函数是许多数学领域不可缺少的工具，在其他科学和工程领域也有广泛的应用。应用科学中许多问题的解决最终可能依赖于复杂分析工具的改进。在这个项目中进行的研究结果可能会导致发现复值函数的新性质。该项目还具有重要的教育和培训方面。研究生、本科生和初级研究人员将积极参与该项目。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，将许多数学家聚集在一起讨论他们的研究和教学。该项目涉及在几个复杂变量和柯西-黎曼几何的广泛领域的许多问题的工作。所考虑的问题也与其他数学领域有联系，包括微分几何、复杂奇点理论、代数几何和经典动力学。更具体地说，PI将继续研究几个复杂变量的刚性问题，以及它们与复杂几何和代数几何的应用和相互作用。他将继续研究几个复变量的等价问题，继续研究复空间中实子流形的全纯壳的复结构，并进一步研究以CR奇点的实子流形为界的列维平面子流形的存在性和正则性问题。此外，他将继续研究弱伪凸域的边界不变量，有限型条件和Bloom猜想，以及具有孤立正态奇点的Stein空间上的规范度量。最后，他将继续研究全纯函数的边界唯一延拓问题和Cauchy-Riemann环境下的横向性问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。