Research and Education in Several Complex Variables

复变量在整个数学中都有使用，并在工程、物理和其他科学学科中有重要的应用。这个项目将澄清和加强对在几个复杂变量中观察到的现象的科学理解。这个项目的结果将导致与多维复分析研究中常用的工具、算子和偏微分方程式相关的重大进展。此外，该项目将帮助研究生和博士后水平的早期职业研究人员的发展。该项目将开展的工作从几个复变量理论中的基本偏微分方程式开始，称为d-bar Neumann问题。要考虑的问题与数学的许多领域相互作用，这项研究的结果将推动偏微分方程式、位势理论、微分几何和算子理论的知识前沿。该项目将进一步研究首席研究员关于d-bar Neumann算子的全局正则性与Diederich-Forness指数之间的联系的研究。此外，首席研究员和联合首席研究员将合作研究d-bar Neumann算子的紧性与复欧氏空间中区域的弱伪凸边界点的局部边界结构之间的关系。最后，首席研究员将与他的博士生合作，研究复杂空间中域的Bergman空间之间的运算符。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。

Complex variables are used throughout mathematics and have important applications to engineering, physics and other scientific disciplines. This project will clarify and enhance the scientific understanding of phenomena observed in several complex variables. The results of this project will lead to significant advances related to commonly used tools, operators, and partial differential equations that originate in the study of multidimensional complex analysis. In addition, the project will aid the development of early career researchers at both the graduate and postdoctoral levels.The work to be carried out in the project initiates from a fundamental partial differential equation in the theory of several complex variables known as the d-bar Neumann problem. The problems to be considered interact with numerous areas of mathematics, and outcomes from this research will push the frontier of knowledge in partial differential equations, potential theory, differential geometry, and operator theory. The project will further the research of the Principal Investigator on the connection between global regularity of the d-bar Neumann operator and the Diederich-Fornaess index. In addition, the Principal Investigator and the co-Principal Investigator will collaborate on a study of the relationship between compactness of the d-bar Neumann operator and the local boundary structure at weakly pseudoconvex boundary points for a domain in complex Euclidean space. Finally, the Principal Investigator, in collaboration with his PhD student, will study operators between Bergman spaces of domains in complex space.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.