CAREER: Geometric Function Theory in Several Complex Variables

职业：多个复变量的几何函数论

• 批准号: 2045104
• 负责人: Ming Xiao
• 金额: $ 42.5万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2045104&HistoricalAwards=false
• 关键词: CAREER; Geometric; Function; Theory; Several

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.

该职业奖将支持PI对几个复杂变量和Cauchy-Riemann几何形状中各种几何和分析问题的调查。该研究的目的是在几个复杂变量中进一步了解几何函数理论，及其与代数几何，复杂几何，复杂几何，动力学系统和物理学方面的联系。该项目将开发新方法，并为研究生和博士后提供有趣的研究主题。 PI将为不同的学术水平以及中学教育者的学生组织一系列教育活动。这包括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课程以及一项本科暑期学校课程。 PI将鼓励来自代表性不足的团体和服务欠缺的学区的人们参加这些活动。这些计划还将为研究生提供教学培训机会。伯格曼内核和公制将在研究中发挥重要作用。特别是，将根据其伯格曼内核和指标来研究开放式复合空间的几何形状。 PI还将对Cauchy-Riemann和Holomorphic映射的规律性和刚性问题进行研究，这些映射自然是在几个复杂变量，复杂的几何形状和算术代数几何形状中产生的。研究中的方法还将结合偏微分方程，代数和差异几何形状的技术，除了复杂的分析外。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的审查标准通过评估来获得支持的。

项目成果

Rigidity of Mappings Between Degenerate and Indefinite Hyperbolic Spaces

• DOI: 10.1007/s12220-022-01080-1
• 发表时间: 2022-11
• 期刊: The Journal of Geometric Analysis
• 作者: Xiaojun Huang;Ming Xiao

Holomorphic mappings between hyperquadrics with positive signature

具有正签名的超二次曲面之间的全纯映射

• DOI: 10.4310/pamq.2022.v18.n2.a11
• 发表时间: 2022
• 期刊: Pure and Applied Mathematics Quarterly
• 影响因子: 0.7
• 作者: Huang, Xiaojun;Xiao, Ming
• 通讯作者: Xiao, Ming

Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains

• DOI: 10.1515/crelle-2022-0029
• 发表时间: 2022-06
• 期刊: Journal für die reine und angewandte Mathematik (Crelles Journal)
• 作者: Ming Xiao

Proper mappings between indefinite hyperbolic spaces and type I classical domains

不定双曲空间与 I 型经典域之间的正确映射

• DOI: 10.1090/tran/8618
• 发表时间: 2022
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Huang, Xiaojun;Lu, Jin;Tang, Xiaomin;Xiao, Ming
• 通讯作者: Xiao, Ming

Kähler-Einstein metrics and obstruction flatness of circle bundles

圆束的克勒-爱因斯坦度量和阻碍平坦度

• DOI: 10.1016/j.matpur.2023.07.003
• 发表时间: 2023
• 期刊: Journal de Mathématiques Pures et Appliquées
• 作者: Ebenfelt, Peter;Xiao, Ming;Xu, Hang
• 通讯作者: Xu, Hang