Applications of Nonpositive Curvature in Several Complex Variables

非正曲率在多复变量中的应用

• 批准号: 1760233
• 负责人: Andrew Zimmer
• 金额: $ 12万
• 依托单位: College of William and Mary
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-10 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1760233&HistoricalAwards=false
• 关键词: Applications; Nonpositive; Curvature; Several; Complex

The main subject of research is complex analysis, which combines complex numbers with the theory of calculus. Complex analysis is a fundamental tool in many applications. In particular, it is used in physics (for instance: studying the flow of air past an airfoil and dispersion relations in optics), engineering (for instance: signal processing and control theory), and computer science (for instance: image processing and quantum computation). Moreover, complex analysis of a single variable is a classical and well understood mathematical subject, but when additional variables are introduced many mysteries remain. The primary aim of the efforts of the PI is to further the theoretical understanding of complex analysis of several variables. The PI will advance the understanding of the behavior of holomorphic maps between bounded domains in higher dimensional complex Euclidean space. In the field of several complex variables, there have been many deep investigations into when holomorphic maps extend continuously to the boundary, the behavior of iterations of holomorphic maps, and the properties of the biholomorphism group of a bounded domain. The standard approach to studying these problems uses methods from partial differential equations and differential geometry. The PI will study these problems using techniques from the theory of non-positively curved metric spaces. This approach is motivated by the great success of geometric group theory, where metric space techniques applied to group theory have lead to many important results. By using metric spaces techniques in several complex variables, the PI will be able to study classes of domains which are typically outside the reach of the standard analytic methods and also make progress on old problems. This part of the activity will enhance knowledge about the biholomorphism group of complex manifolds, connections between the boundary of a domain and its complex geometry, the iterations of holomorphic maps, continuous extensions of holomorphic maps, realizations of Hermitian symmetric spaces, and the complex geometry of certain smooth quasi-projective algebraic varieties.

主要研究课题是复数分析，将复数与微积分理论相结合。复杂分析是许多应用中的基本工具。特别是，它用于物理学（例如：研究经过机翼的空气流动和光学中的色散关系）、工程（例如：信号处理和控制理论）和计算机科学（例如：图像处理和量子计算）。此外，单个变量的复杂分析是一门经典且易于理解的数学学科，但当引入其他变量时，仍然存在许多谜团。 PI 工作的主要目的是进一步加深对多个变量的复杂分析的理论理解。 PI 将促进对高维复杂欧几里德空间中有界域之间的全纯映射行为的理解。在多复变量领域，人们对全纯映射连续延伸到边界时、全纯映射的迭代行为以及有界域双全纯群的性质进行了许多深入的研究。研究这些问题的标准方法使用偏微分方程和微分几何的方法。 PI 将使用非正弯曲度量空间理论的技术来研究这些问题。这种方法是由几何群论的巨大成功推动的，其中度量空间技术应用于群论已经产生了许多重要的结果。通过在多个复杂变量中使用度量空间技术，PI 将能够研究通常超出标准分析方法范围的领域类别，并在老问题上取得进展。这部分活动将增强有关复流形双全纯群、域边界与其复几何之间的联系、全纯映射的迭代、全纯映射的连续扩展、埃尔米特对称空间的实现以及某些光滑准射影代数簇的复几何的知识。

项目成果

Smoothly bounded domains covering finite volume manifolds

• DOI: 10.4310/jdg/1631124346
• 发表时间: 2018-02
• 期刊: Journal of Differential Geometry
• 影响因子: 2.5
• 作者: Andrew M. Zimmer

The automorphism group and limit set of a bounded domain II: the convex case

有界域的自同构群和极限集 II：凸情况

• DOI: 10.1112/jlms.12435
• 发表时间: 2021
• 期刊: Journal of the London Mathematical Society
• 作者: Zimmer, Andrew

Two boundary rigidity results for holomorphic maps

全纯贴图的两个边界刚度结果

• DOI: 10.1353/ajm.2022.0002
• 发表时间: 2022
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: Zimmer, Andrew

Asymptotic behavior of orbits of holomorphic semigroups

全纯半群轨道的渐近行为

• DOI: 10.1016/j.matpur.2019.05.005
• 发表时间: 2020
• 期刊: Journal de Mathématiques Pures et Appliquées
• 作者: Bracci, Filippo;Contreras, Manuel D.;Díaz-Madrigal, Santiago;Gaussier, Hervé;Zimmer, Andrew
• 通讯作者: Zimmer, Andrew

Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents

• DOI: 10.1007/s00208-018-1715-7
• 发表时间: 2017-03
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Andrew M. Zimmer