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This book focuses on gathering the numerous properties and many different connections with various topics in geometric function theory that quasidisks possess. A quasidisk is the image of a disk under a quasiconformal mapping of the Riemann sphere. In 1981 Frederick W. Gehring gave a short course of six lectures on this topic in Montreal and his lecture notes “Characteristic Properties of Quasidisks” were published by the University Press of the University of Montreal. The notes became quite popular and within the next decade the number of characterizing properties of quasidisks and their ramifications increased tremendously. In the late 1990s Gehring and Hag decided to write an expanded version of the Montreal notes. At three times the size of the original notes, it turned into much more than just an extended version. New topics include two-sided criteria. The text will be a valuable resource for current and future researchers in various branches of analysis and geometry, and with its clear and elegant exposition the book can also serve as a text for a graduate course on selected topics in function theory. Frederick W. Gehring (1925–2012) was a leading figure in the theory of quasiconformal mappings for over fifty years. He received numerous awards and shared his passion for mathematics generously by mentoring twenty-nine Ph.D. students and more than forty postdoctoral fellows. Kari Hag received her Ph.D. under Gehring's direction in 1972 and worked with him on the present text for more than a decade.

This work begins with the presentation of generalizations of the classical Herglotz Representation Theorem for holomorphic functions with positive real part on the unit disc to functions with positive real part defined on multiply-connected domains. The generalized Herglotz kernels that appear in these representation theorems are then exploited to evolve new conditions for spectral set and rational dilation conditions over multiply-connected domains. These conditions form the basis for the theoretical development of a computational procedure for probing a well-known unsolved problem in operator theory, the so called rational dilation conjecture. Arbitrary precision algorithms for computing the Herglotz kernels on circled domains are presented and analyzed. These algorithms permit an effective implementation of the computational procedure which results in a machine generated counterexample to the rational dilation conjecture.

For over 70 years, the Bieberbach conjecture has intrigued the mathematical world. Many students of mathematics, who have had a first course in function theory, have tried their hand at a proof. But many have invested fruitless years of carefully manipulating inequalities in an attempt to establish the correct bound. In 1977, Louis de Branges of Purdue University took up the challenge of this famous unsolved problem, but in his case the outcome was different. He will be recognized as the mathematician who proved Bieberbach's conjecture. And more importantly, his method came from totally unexpected sources: operator theory and special functions. This book, based on the Symposium on the Occasion of the Proof, tells the story behind this fascinating proof and offers insight into the nature of the conjecture, its history and its proof. A special and unusual feature of the book is the enlightened personal accounts of the people involved in the exciting events surrounding the proof. Especially attractive are the photographs of mathematicians who have made significant contributions to univalent functions, the area of complex analysis which provides the setting for the Bieberbach conjecture. Research mathematicians, especially analysts, are sure to enjoy the articles in this volume. Most articles require only a basic knowledge of real and complex analysis. The survey articles are accessible to non-specialists, and the personal accounts of all who have played a part in this important discovery will fascinate any reader. “The remarks by de Branges himself about the discovery of his proof should be read by all young mathematicians. He describes the difficulty he had in convincing the experts in the field that a mathematician, whose work was considered to lie in an entirely different area, had actually proved a problem of such long standing. When a mathematician is sure that he has the solution of a problem, he must persist until he convinces others or is actually proved wrong.” (Prepublication comments by James A. Hummel, The University of Maryland, College Park.)