Function Theory of Several Complex Variables

这个项目涉及对复数和复函数的理解。复数和复变函数在数学的许多领域都是不可或缺的工具，并在科学和工程的其他领域有着深刻的应用。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。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特性。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关于几个复变量和复几何的国际会议，汇集许多数学家讨论他们的研究和教学。该项目涉及多个复变量和柯西-黎曼几何的广泛领域的一些问题的工作。所考虑的问题也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将继续他的刚性问题的调查，在几个复杂的变量，沿着与他们的应用和相互作用与复杂的几何和代数几何。他将继续他的研究等价问题在几个复杂的变量，追求他正在进行的研究复杂的结构的全纯船体的一个真实的子流形在一个复杂的空间，并进一步他的调查存在和规律性问题的列维平坦子流形有界的真实的子流形与CR奇点。此外，他将继续他的工作边界不变量的弱伪凸域，有限型条件和布卢姆猜想，规范度量与孤立的正常奇异斯坦空间。最后，他将继续研究全纯函数的边界唯一延拓问题，以及柯西-黎曼背景下的横截性问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值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.