ABSTRACT

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.

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