Mathematical Surveys and Monographs 1986; 218 pp; softcover Volume: 21 Reprint/Revision History: reprinted 1987 ISBN10: 0821815210 ISBN13: 9780821815212 List Price: US$70 Member Price: US$56 Order Code: SURV/21
 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 nonspecialists, 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.) Table of Contents Mathematical Papers  L. V. Ahlfors  Classical analysis: present and future
 R. Askey and G. Gasper  Inequalities for polynomials
 A. Beurling  On interpolation, Blaschke products, and balayage of measures
 L. de Branges  Powers of Riemann mapping functions
 J. Dieudonné  300 years of analyticity
 P. R. Garabedian  Problems in mathematical physics connected with the Bieberbach conjecture
 D. H. Hamilton  Extremal methods
 J. A. Jenkins  The method of the extremal metric
 P. W. Jones  Some problems in complex analysis
 I. M. Milin  Comments on the proof of the conjecture on logarithmic coefficients
 N. K. Nikol'skiĭ and V. I. Vasyunin  Notes on two function models
 C. Pommerenke  The growth of the derivative of a univalent function
 D. Sarason  Shiftinvariant subspaces from the Brangesian point of view
 S. W. Semmes  The Cauchy integral, chordarc curves, and quasiconformal mappings
 W. P. Thurston  Zippers and univalent functions
Accounts  L. de Branges  The story of the verification of the Bieberbach conjecture
 W. Gautschi  Reminiscences of my involvement in de Branges's proof of the Bieberbach conjecture
 R. Askey  My reaction to de Branges's proof of the Bieberbach conjecture
 W. Fuchs  Poem
