
Preface  Preview Material  Table of Contents  Index  Supplementary Material 
Graduate Studies in Mathematics 2014; 384 pp; hardcover Volume: 154 ISBN10: 0821898477 ISBN13: 9780821898475 List Price: US$79 Member Price: US$63.20 Order Code: GSM/154 See also: Riemann Surfaces by Way of Complex Analytic Geometry  Dror Varolin A Scrapbook of Complex Curve Theory: Second Edition  C Herbert Clemens Complex Made Simple  David C Ullrich  Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics. Schlag's treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of Riemann surfaces. The book begins with an exposition of the basic theory of holomorphic functions of one complex variable. The first two chapters constitute a fairly rapid, but comprehensive course in complex analysis. The third chapter is devoted to the study of harmonic functions on the disk and the halfplane, with an emphasis on the Dirichlet problem. Starting with the fourth chapter, the theory of Riemann surfaces is developed in some detail and with complete rigor. From the beginning, the geometric aspects are emphasized and classical topics such as elliptic functions and elliptic integrals are presented as illustrations of the abstract theory. The special role of compact Riemann surfaces is explained, and their connection with algebraic equations is established. The book concludes with three chapters devoted to three major results: the Hodge decomposition theorem, the RiemannRoch theorem, and the uniformization theorem. These chapters present the core technical apparatus of Riemann surface theory at this level. This text is intended as a detailed, yet fastpaced intermediate introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics, including geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. More than seventy figures serve to illustrate concepts and ideas, and the many problems at the end of each chapter give the reader ample opportunity for practice and independent study. Request an examination or desk copy. Readership Graduate students interested in complex analysis, conformal geometry, Riemann surfaces, uniformization, harmonic functions, differential forms on Riemann surfaces, and the RiemannRoch theorem. 


AMS Home 
Comments: webmaster@ams.org © Copyright 2014, American Mathematical Society Privacy Statement 