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对多复变量和Cauchy-Riemann几何中各种几何和分析问题的研究。研究的目的是进一步了解几何函数理论在几个复杂的变量，以及它与代数几何，复几何，动力系统和物理方面的连接。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和中学教育工作者举办一系列教育活动。这包括高中学生的阅读和研究计划，高中教师的学习和教学计划，以及本科生暑期学校计划。PI将鼓励代表性不足的群体和服务不足的学区的人参与这些活动。这些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。特别是，开放的复杂空间的几何将在其伯格曼内核和度量方面进行研究。PI还将研究Cauchy-Riemann和全纯映射的正则性和刚性问题，这些问题在多复变量，复几何和算术代数几何中自然出现。研究方法将结合偏微分方程、代数和微分几何技术以及复杂的分析。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。

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

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

A high-order Hopf lemma for mappings into classical domains and applications

用于映射到经典领域和应用程序的高阶 Hopf 引理

• DOI: 10.4310/cag.2021.v29.n8.a7
• 发表时间: 2021
• 期刊: Communications in Analysis and Geometry
• 影响因子: 0.7
• 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