Five Lectures in Complex Analysis (PDF)

This volume contains state-of-the-art survey papers in complex analysis based on lectures given at the Second Winter School on Complex Analysis and Operator Theory held in February 2008 at the University of Sevilla, Sevilla, Spain. Complex analysis is one of the most classical branches of mathematical analysis and is closely related to many other areas of mathematics, including operator theory, harmonic analysis, probability theory, functional analysis and dynamical systems. Undoubtedly, the interplay among all these branches gives rise to very beautiful and deep results in complex analysis and its neighboring fields. This interdisciplinary aspect of complex analysis is the central topic of this volume. This book collects the latest advances in five significant areas of rapid development in complex analysis. The papers are: Local holomorphic dynamics of diffeomorphisms in dimension one, by F. Bracci, Nonpositive curvature and complex analysis, by S. M. Buckley, Virasoro algebra and dynamics in the space of univalent functions, by I. Markina and A. Vasil'ev, Composition operators $\heartsuit$ Toeplitz operators, by J. H. Shapiro, and Two applications of the Bergman spaces techniques, by S. Shimorin. The papers are aimed, in particular, at graduate students with some experience in basic complex analysis. They might also serve as introductions for general researchers in mathematical analysis who may be interested in the specific areas addressed by the authors. Indeed, the contributions can be considered as up-to-the-minute reports on the current state of the fields, each of them including many recent results which may be difficult to find in the literature.