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Product Code:  SURV/21 
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Book DetailsMathematical Surveys and MonographsVolume: 21; 1986; 218 ppMSC: Primary 30;
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 [ MR MR875226 ]

Lars V. Ahlfors II  1. Classical analysis: Present and future [ MR 875227 ]

Richard Askey II and George Gasper  2. Inequalities for polynomials [ MR 875228 ]

Arne Beurling II  3. On interpolation, Blaschke products, and balayage of measures [ MR 875229 ]

Louis de Branges II  4. Powers of Riemann mapping functions [ MR 875230 ]

Jean Dieudonné II  5. 300 years of analyticity [ MR 875231 ]

Paul R. Garabedian II  6. Problems in mathematical physics connected with the Bieberbach conjecture [ MR 875232 ]

D. H. Hamilton II  7. Extremal methods [ MR 875233 ]

James A. Jenkins II  8. The method of the extremal metric [ MR 875234 ]

Peter W. Jones II  9. Some problems in complex analysis [ MR 875235 ]

I. M. Milin II  10. Comments on the proof of the conjecture on logarithmic coefficients [ MR 875236 ]

N. K. Nikol’skiĭ II and V. I. Vasyunin  11. Notes on two function models [ MR 875237 ]

Christian Pommerenke II  12. The growth of the derivative of a univalent function [ MR 875238 ]

Donald Sarason II  13. Shiftinvariant subspaces from the Brangesian point of view [ MR 875239 ]

Stephen W. Semmes II  14. The Cauchy integral, chordarc curves, and quasiconformal mappings [ MR 875240 ]

William P. Thurston II  15. Zippers and univalent functions [ MR 875241 ]

Personal accounts [ MR MR875226 ]

Louis de Branges II  16. The story of the verification of the Bieberbach conjecture [ MR 875242 ]

Walter Gautschi II  17. Reminiscences of my involvement in de Branges’s proof of the Bieberbach conjecture [ MR 875243 ]

Richard Askey II  18. My reaction to de Branges’s proof of the Bieberbach conjecture [ MR 875244 ]

Wolfgang Fuchs II  19. Poem [ MR 875245 ]


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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.)

Mathematical papers [ MR MR875226 ]

Lars V. Ahlfors II  1. Classical analysis: Present and future [ MR 875227 ]

Richard Askey II and George Gasper  2. Inequalities for polynomials [ MR 875228 ]

Arne Beurling II  3. On interpolation, Blaschke products, and balayage of measures [ MR 875229 ]

Louis de Branges II  4. Powers of Riemann mapping functions [ MR 875230 ]

Jean Dieudonné II  5. 300 years of analyticity [ MR 875231 ]

Paul R. Garabedian II  6. Problems in mathematical physics connected with the Bieberbach conjecture [ MR 875232 ]

D. H. Hamilton II  7. Extremal methods [ MR 875233 ]

James A. Jenkins II  8. The method of the extremal metric [ MR 875234 ]

Peter W. Jones II  9. Some problems in complex analysis [ MR 875235 ]

I. M. Milin II  10. Comments on the proof of the conjecture on logarithmic coefficients [ MR 875236 ]

N. K. Nikol’skiĭ II and V. I. Vasyunin  11. Notes on two function models [ MR 875237 ]

Christian Pommerenke II  12. The growth of the derivative of a univalent function [ MR 875238 ]

Donald Sarason II  13. Shiftinvariant subspaces from the Brangesian point of view [ MR 875239 ]

Stephen W. Semmes II  14. The Cauchy integral, chordarc curves, and quasiconformal mappings [ MR 875240 ]

William P. Thurston II  15. Zippers and univalent functions [ MR 875241 ]

Personal accounts [ MR MR875226 ]

Louis de Branges II  16. The story of the verification of the Bieberbach conjecture [ MR 875242 ]

Walter Gautschi II  17. Reminiscences of my involvement in de Branges’s proof of the Bieberbach conjecture [ MR 875243 ]

Richard Askey II  18. My reaction to de Branges’s proof of the Bieberbach conjecture [ MR 875244 ]

Wolfgang Fuchs II  19. Poem [ MR 875245 ]