Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including: invariant geometry, the Bergman metric, the automorphism groups of domains, extremal length, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous Cauchy-Riemann equations, and the corona problem.The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.Containing an extensive bibliography of both monographs and research papers and a thorough index, the book is methodically designed: the individual chapters contain a rich collection of exercises, examples, and illustrations. Seeking to capture the imagination of both advanced undergraduate and graduate students with a basic background in complex analysis-and also to spark the interest of seasoned workers in the field-the book imparts a solid education both in complex analysis and in how modern mathematics works.

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