CAREER: Geometric Function Theory in Several Complex Variables

该CAREER奖将支持PI在多个复杂变量和Cauchy-Riemann几何中的各种几何和解析问题的研究。本研究的目的是进一步了解几种复杂变量的几何函数理论，以及它与代数几何、复杂几何、动力系统和物理等方面的联系。该项目将开发新的方法，并为研究生和博士后提供有趣的研究课题。PI将为不同学术水平的学生以及中学教育工作者组织一系列教育活动。这包括一个针对高中生的阅读和研究计划，一个针对高中教师的学习和教学计划，以及一个本科暑期学校计划。PI将鼓励来自代表性不足的群体和服务不足的学区的人参与这些活动。该项目还将为研究生提供教学培训机会。Bergman核和度量将在研究中发挥重要作用。特别地，开放复空间的几何将根据它们的Bergman核和度量来研究。PI还将研究在一些复杂变量、复杂几何和算术代数几何中自然出现的柯西-黎曼和全纯映射的正则性和刚性问题。研究方法将结合偏微分方程、代数和微分几何的技术，以及复杂分析。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。

This CAREER award will support the PI's investigations of various geometric and analytic problems in several complex variables and Cauchy-Riemann geometry. The objective of the research is to further the present understanding of geometric function theory in several complex variables, as well as its connections with aspects of algebraic geometry, complex geometry, dynamical systems and physics. The project will develop new methods and provide interesting research topics for graduate students and postdocs. The PI will organize a series of educational activities for students of different academic levels, as well as secondary school educators. This includes a reading and research program for high school students, a learning and teaching program for high school teachers, and an undergraduate summer school program. The PI will encourage the participation of people from underrepresented groups and underserved school districts into these activities. The programs will also provide pedagogical training opportunities for graduate students. The Bergman kernel and metric will play a prominent role in the research. In particular, the geometry of open complex spaces will be investigated in terms of their Bergman kernels and metrics. The PI will also conduct research on the regularity and rigidity problems of Cauchy-Riemann and holomorphic mappings that naturally arise in several complex variables, complex geometry, and arithmetical algebraic geometry. The methods in the research will incorporate techniques from partial differential equations, algebra, and differential geometry, in addition to complex analysis.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.