Function Theory of Several Complex Variables

这个项目涉及对复数和复数函数的理解。复数和复变函数是许多数学领域中不可缺少的工具，在科学和工程的其他领域有着广泛的应用。应用科学中许多问题的解决最终可能取决于复杂分析工具的改进。本项目将开展的研究结果可能会导致复值函数的新性质的发现。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，首席研究人员将继续组织关于几个复变量和复杂几何的国际会议，聚集许多数学家讨论他们的研究和教学。这个项目涉及到在几个复变量和柯西-黎曼几何的广泛领域中的一些问题的工作。正在考虑的问题还与其他数学领域有关，包括微分几何、复奇点理论、代数几何和经典动力学。更具体地说，PI将继续他对几个复变量的刚性问题的研究，以及它们的应用和与复几何和代数几何的相互作用。他将继续研究多个复变量的等价问题，继续他正在进行的关于复空间中实子流形的全纯壳的复结构的研究，并进一步研究由具有CR奇点的实子流形有界的Levi平坦子流形的存在性和正则性问题。此外，他还将继续研究弱伪凸域的边界不变量、有限类型条件和Bloom猜想，以及具有孤立正规奇点的Stein空间上的正则度量。最后，他将继续研究全纯函数的边界唯一连续性问题，并在柯西-黎曼环境下研究横截性问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.