**Louis de Branges** is a French-born American mathematician who is best known for proving the long-standing Bieberbach conjecture.

- Louis began his education in Louveciennes, in the western suburbs of Paris, in 1937.
- Despite the war Louis continued his education at Louveciennes spending his third year at the school although by this time the school buildings could not be used as they had been taken over as a German military headquarters and the de Branges' home was occupied by German soldiers.
- In his second year de Branges took Walter Rudin's course on the 'Principles of Mathematical Analysis'.
- By his third year he had made a decision that he would try to prove the Riemann hypothesis, an aim which would dominate his life from that point on.
- He attended a symposium on harmonic analysis at Cornell University in the summer 1956 and a problem arose in a lecture on the spectral theory of unbounded functions given by Szolem Mandelbrojt.
- He was awarded a Ph.D. from Cornell University for this thesis in 1957.
- After the award of his doctorate, de Branges was appointed as an Assistant Professor of Mathematics at Lafayette College in Easton, Pennsylvania.
- He spent two years at Lafayette, leaving in 1959 to spend the academic year 1959-60 at the Institute for Advanced Study at Princeton.
- Let us now look at the remarkable mathematics de Branges has produced.
- After completing his doctorate, de Branges worked on Hilbert spaces of entire functions.
- The treatment of entire functions developed in these and subsequent papers by de Branges was published in his 326-page book Hilbert spaces of entire functions in 1968.
- There are occasions when the phrase "it is easily seen that" hides an incorrect statement.
- Louis de Branges is a courageous exception; his originality is his own.
- However, de Branges solved one of the most important conjectures in mathematics in 1984, namely he solved the Bieberbach conjecture which, as a result, is now called 'de Branges' theorem'.
- For this achievement he was awarded the Ostrowski Prize, presented to him on 4 May 1990 at the Mathematical Institute of the University of Basel.
- he began investigating the question of whether every bounded linear operator on Hilbert space has a non-trivial invariant subspace, and also worked on the Riemann hypothesis.
- His Hilbert space theory has contributed substantially to the understanding of these and other problems.
- He was also awarded the American Mathematical Society's 1994 Steele Prize.
- The Steele Prize is awarded to him for the paper "A proof of the Bieberbach conjecture" published in 'Acta Mathematica' in 1985.
- The classical ingredients of the proof, the Loewner differential equation and the inequalities conjectured by Robertson and Milin, as well as the Askey-Gasper inequalities from the theory of special functions, are clearly described in the volume 'The Bieberbach Conjecture' (published by the American Mathematical Society).
- So is the generous reception of the Leningrad mathematicians to the efforts of de Branges to explain it and their help in the composition of the eminently readable 'Acta' paper.
- The Milin inequality was known to imply the Bieberbach conjecture, and Loewner had used his techniques in the 1920s to deal with the third coefficient.
- For de Branges it was of capital importance that, in contrast to the Bieberbach conjecture itself, the Milin and Robertson conjectures were quadratic and thus statements about spaces of square-integrable analytic functions.
- de Branges constructed the necessary coefficients from scratch, reducing the verification of the Milin conjecture (and thus of the Bieberbach conjecture) for a given integer nnn to a statement that was almost immediate for very small nnn, that could be verified numerically for small nnn, yielding many new cases of the conjecture, and that ultimately revealed itself to be an inequality established several years earlier by Askey and Gasper.
- Although the mathematical community does not attach the same importance to the general functional-analytic principles that led to them as the author does, it is well to remember when recognising his achievement in proving the Bieberbach conjecture that for de Branges its appeal, like that of other conjectures from classical function theory, is as a touchstone for his contributions to interpolation theory and spaces of square-summable analytic functions.
- One is made to the American Mathematical Society for its continued endorsement of research related to the Bieberbach conjecture.
- The American Mathematical Society has earned a reputation as the world's foremost leader in fundamental scientific research.
- Another acknowledgement is due to Ludwig Bieberbach as a founder of that branch of twentieth century mathematics which has come to be known as functional analysis.
- The issue which divided Bieberbach from these illustrious colleagues is relevant to the present day because it concerns the teaching of mathematics.
- Bieberbach originated the widely held current view that mathematical teaching is not second to mathematical research.
- Progress has also been made towards the initial objective of a proof of the Riemann hypothesis.
- At the International Congress of Mathematicians held in Berkeley, California, in August 1986, de Branges was an invited plenary speaker and gave the address Underlying concepts in the proof of the Bieberbach conjecture.
- Now we mentioned earlier that de Branges' life has been dominated by his aim to prove the Riemann hypothesis and he indicated in the above quote that his proof of the Bieberbach conjecture is, in many ways, a consequence of the work he was doing attacking the Riemann hypothesis.
- Most mathematicians doubt that de Branges' proof is correct but, of course, even if it is not correct it is not impossible that the ideas that it contains could eventually lead to a correct proof.
- In December 2008, de Branges posted a paper on his website which claimed to prove the invariant subspace conjecture of which he had given an incorrect proof in 1964.
- A person with a fixed idea will always find some way of convincing himself in the end that he is right.
- Louis de Branges has committed a lot of mistakes in his life.
- Unfortunately, his reputation is somewhat tainted by several claims he made in the past, whose proofs eventually collapsed.
- Mathematics is always considered to be a young man's game, so it would be most interesting if a 70-year-old mathematician were to prove the Riemann Hypothesis, which has been considered to be the Holy Grail of mathematics for about a hundred years.

Born 21 August 1932, Neuilly-sur-Seine, Paris, France.

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**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive