Applications of Nonpositive Curvature in Several Complex Variables

非正曲率在多复变量中的应用

• 批准号: 2104381
• 负责人: Andrew Zimmer
• 金额: $ 1.86万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-17 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104381&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.

研究的主要课题是复分析，它将复数与微积分理论相结合。复分析是许多应用中的基本工具。特别是，它被用于物理学（例如：研究空气流过机翼的流动和光学中的色散关系），工程学（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，单一变量的复分析是一个经典的和很好理解的数学主题，但当引入额外的变量时，许多谜团仍然存在。PI努力的主要目的是进一步理解多变量复分析的理论。PI将推进对高维复欧氏空间中有界域之间全纯映射行为的理解。在多复变域中，全纯映射何时连续延拓到边界，全纯映射的迭代行为，有界域的双全纯群的性质等都有许多深入的研究。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中的技术来研究这些问题。这种方法的动机是几何群论的巨大成功，其中度量空间技术应用于群论导致了许多重要的结果。通过在多个复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域类，并在旧问题上取得进展。这部分的活动将提高知识的biholomorphism群的复杂的流形，一个域的边界和它的复杂的几何形状，全纯映射的迭代，连续扩展的全纯映射，实现厄米特对称空间，和复杂的几何形状的某些光滑的准投射代数簇。

项目成果

Unbounded visibility domains, the end compactification, and applications

无界可见域、最终紧凑化和应用

• DOI: 10.1090/tran/8944
• 发表时间: 2023
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Bharali, Gautam;Zimmer, Andrew
• 通讯作者: Zimmer, Andrew

Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

几何有限 Fuchsian 群的尖点 Hitchin 表示和 Anosov 表示

• DOI: 10.1016/j.aim.2022.108439
• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Canary, Richard;Zhang, Tengren;Zimmer, Andrew
• 通讯作者: Zimmer, Andrew

Smoothly bounded domains covering compact manifolds

覆盖紧凑流形的平滑边界域

• DOI: 10.1512/iumj.2021.70.9380
• 发表时间: 2021
• 期刊: Indiana University Mathematics Journal
• 影响因子: 1.1
• 作者: Zimmer, Andrew

Hankel Operators on Domains with Bounded Intrinsic Geometry

有界本征几何域上的 Hankel 算子

• DOI: 10.1007/s12220-023-01231-y
• 发表时间: 2023
• 期刊: The Journal of Geometric Analysis
• 作者: Zimmer, Andrew

Convex cocompact actions of relatively hyperbolic groups

相对双曲群的凸协紧作用

• DOI: 10.2140/gt.2023.27.417
• 发表时间: 2023
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Islam, Mitul;Zimmer, Andrew
• 通讯作者: Zimmer, Andrew