Applications of Nonpositive Curvature in Several Complex Variables

研究的主要课题是复分析，它将复数与微积分理论相结合。复杂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：研究气流经过机翼和色散关系在光学），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，单变量的复杂分析是一个经典的、很容易理解的数学主题，但是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从理论上理解多变量的复杂分析。PI将促进对高维复欧几里德空间中有界域之间全纯映射行为的理解。在复数变量领域，对全纯映射连续扩展到边界时，全纯映射的迭代行为，有界域上的生物全纯群的性质等问题进行了深入的研究。研究这些问题的标准方法是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已经导致了许多重要的结果。通过在几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。这部分活动将增强对复流形的生物全纯群、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、厄米对称空间的实现以及某些光滑拟射影代数变体的复几何的认识。

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.