CAREER: Geometric Function Theory in Several Complex Variables

这一职业奖将支持PI对多个复变量和Cauchy-Riemann几何中的各种几何和解析问题的研究。这项研究的目的是加深对多元几何函数论的理解，以及它与代数几何、复几何、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主题。国际学生联合会将为不同学历的学生和中学教育工作者举办一系列教育活动。这包括一个面向高中生的阅读和研究计划，一个面向高中教师的学习和教学计划，以及一个本科生暑期学校计划。国际和平协会将鼓励来自代表人数不足的群体和服务不足的学区的人参加这些活动。这些项目还将为研究生提供教学培训机会。伯格曼核和度量将在这项研究中发挥重要作用。特别是，将根据其Bergman核和度量来研究开放复杂空间的几何。PI还将研究柯西-黎曼映射和全纯映射的正则性和刚性问题，这些映射自然地出现在多个复变量、复几何和算术代数几何中。研究中的方法将结合偏微分方程、代数和微分几何的技术，以及复杂的分析。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 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.