Applications of Nonpositive Curvature in Several Complex Variables

研究的主要内容是复数分析，它把复数和微积分理论结合起来。在许多应用中，复杂分析是一种基本工具。特别是，它被用于物理学(例如：研究空气通过机翼的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，对单一变量的复杂分析是一个经典的、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努力的主要目的是加深对几个变量的复杂分析的理论理解。PI将促进对高维复欧氏空间中有界域之间的全纯映射行为的理解。在多复变领域中，关于全纯映射何时连续扩张到边界、全纯映射的迭代行为以及有界域的双全纯群的性质已经有了许多深入的研究。研究这些问题的标准方法是使用偏微分方程式和微分几何的方法。PI将使用非正曲线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区域类别，并在旧问题上取得进展。这部分活动将加强关于复流形的双全纯群、区域边界与其复几何之间的联系、全纯映射的迭代、全纯映射的连续扩张、厄米特对称空间的实现以及某些光滑拟投射代数簇的复几何的知识。

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.