**Louis de Branges**was born in Neuilly-sur-Seine, a residential suburb of Paris lying to the northwest of the centre of the city. His father, also named Louis de Branges, had been born into a French-American family living in Wayne, a suburb of Philadelphia in the United States. On 8 August 1931 in Ontario, Canada, he married Diane McDonald, the daughter of Ellice McDonald, a surgeon and professor at the University of Pennsylvania. Shortly after Louis de Branges, who was twenty-five years old, sailed with his wife to France where they settled into an apartment in the Square du Roule, Paris. He obtained a position with the Compagnie Générale Transatlantique, a major French shipping company. Louis and Diane had three children, Louis (the subject of this biography), Elise (born 1935), and Eéonore (born 1938). One of the perks of working for a shipping company was a free trip to New York each year and this trip was always made by Diane and children so Louis (the subject of this biography), although brought up in France, was in the United States when a child and met his American relations.

Louis began his education in Louveciennes, in the western suburbs of Paris, in 1937. In 1939, when Louis was seven years old, World War II began and his father enlisted in the French Army. In May 1940 the German army attacked France and quickly forced the French and British armies to retreat to Dunkirk where they were evacuated to England. Louis's father was among the French troops evacuated but he soon returned by ship to France. 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 1941 Diane's father persuaded his daughter to return to the United States and she went with her children, taking a train to Lisbon from where they managed to get a passage on a ship sailing to New York. Louis's father remained in France.

The family settled first in Rehoboth Beach, Delaware but then moved to a house near Wilmington and de Branges entered Saint Andrew's School in Middletown, Delaware. He writes [2]:-

At Saint Andrew's School, de Branges studied hard aware that his grandmother was finding life harder because of the dependence of his mother and her children. He solved hundreds of algebra problems on his own but he was driven to work hard on a problem given to him by Irénée du Pont, a wealthy friend of de Branges' grandfather. The problem was to find integersThe transition to English as a language seems to have stimulated mathematical ability. ... The end of childhood was caused by two events when I was twelve. I entered the second form at Saint Andrew's School as the cottage in Rehoboth was sold. My new home was the house near Wilmington which my grandparents were building when I came from France. My grandmother replaced my mother as the central person in family life.

*a*,

*b*and

*c*such that

This problem, known as the Lagrange problem, took de Branges over a year to solve, but during that time he learnt a lot of mathematics [2]:-a^{3}+b^{3}= 22c^{3}.

It was de Branges' grandfather who decided that he should go to university, but the choice of the Massachusetts Institute of Technology was probably made at Irénée du Pont's suggestion. After graduating from Saint Andrew's School, de Branges took the entrance examinations and began his studies in Boston in September 1949 [2]:-The Lagrange problem taught me to work without expecting reward and yet believing that I would benefit from the work done. The Lagrange problem also taught me to search for mathematical information from non-mathematical sources.

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. After graduating from the Massachusetts Institute of Technology in 1953, de Branges went to Cornell University to undertake graduate studies for a doctorate [2]:-I treated my undergraduate studies as if I were a graduate student. George Thomas was writing a text on the calculus and analytic geometry which was tested on the incoming freshman class. Professor Thomas himself taught the section in which I was placed. I worked through the exercises for all four semesters and was exempted from the remaining three semesters by a proficiency examination. Professor Thomas was pleased by my reading of his untested lecture notes. ... I was freed in the second semester to take a graduate course in linear analysis taught by Witold Hurewicz. ... In the summer break I read the recently published Lectures on Classical Differential Geometry by Dirk Struik. When my knowledge was tested in a proficiency examination, Professor Struik gave me more than a perfect score since I had to correct the statement of one of the problems before solving it.

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 encouraged to study this problem by Harry Pollard, and Wolfgang Fuchs advised him on the relevant literature [2]:-... in graduate years at Cornell University, ... I obtained a teaching assistantship on the recommendation of George Thomas. I approached graduate studies as if I were a postdoctoral fellow.

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. He was appointed for the year 1960-61 as a lecturer at Bryn Mawr College following which he spent the year 1961-62 at the Courant Institute of Mathematical Sciences in New York. He was appointed as an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana, in 1962 and promoted to Professor in the following year. He has been on the Faculty at Purdue from that time onwards. In 1963-66 he was an Alfred P Sloan Foundation Fellow, and in 1967-68 a Guggenheim Fellow. He is currently Edward C Elliot Distinguished Professor of Mathematics at Purdue University.My thesis, 'Local operators on Fourier transforms', clarifies the appearance of entire functions in the spectral theory of unbounded functions.

In [7] de Branges spoke of his personal life, in particular about his first marriage which led to divorce:-

De Branges remarried on 17 December 1980 to Tatiana Jakimov. They have one son Konstantin.I'd married a student from Bryn Mawr College, and all of a sudden she just left, asking for a very substantial amount of money which I didn't in any way contest. And then staying around in Lafayette for about ten years, that greatly created a circle of opposition within the community, because, you see, I was a person that was seen as being in the wrong by my colleagues, and also by the community. The divorce was seen as a criticism of myself, of my performance.

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. After publishing the results from his thesis in 1958, he published five papers in 1959, namely: *The Stone-Weierstrass theorem*; *Some mean squares of entire functions*; *Some Hilbert spaces of entire functions*; *The Bernstein problem*; and *The a-local operator problem*. Following this, he published five papers entitled *Some Hilbert spaces of entire functions* which appeared in 1960-62. 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.

Most mathematicians will from time to time have errors in their work, usually in the form of missing steps in a proof. There are occasions when the phrase "it is easily seen that" hides an incorrect statement. Some would suggest that de Branges has made more mistakes than most, and if this is so then it is probably because of his remarkably innovative approach to mathematics [1]:-

Of these errors, de Branges himself said [7]:-It was observed by Rolf Schwarzenberger that mathematicians strive to be original but seldom in an original way. 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. The citation reads [3]:-The first case in which I made an error was in proving the existence of invariant subspaces for continuous transformations in Hilbert spaces. This was something that happened in1964, and I declared something to be true which I was not able to substantiate. And the fact that I did that destroyed my career. My colleagues have never forgiven it.

He was also awarded the American Mathematical Society's 1994 Steele Prize. The citation gives details of his achievement [1]:-Louis de Branges of Purdue University has received the first Ostrowski Prize for developing powerful Hilbert space methods which led him to a surprising proof of the Bieberbach Conjecture on power series for conformal mappings. ... After receiving his Ph.D. ... 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. However, his greatest accomplishment was the1984proof of the Bieberbach Conjecture, which surprised the mathematical world accustomed to small steps forward. In addition to that proof, he obtained certain more general results concerning conformal maps. His Hilbert space theory has contributed substantially to the understanding of these and other problems.

In his reply, de Branges spoke about his proof of the Bieberbach conjecture and also about some later work [1]:-The Bieberbach Conjecture, formulated in1916and the object of heroic efforts over the years by many outstanding mathematicians, was proved by de Branges in1984. The Steele Prize is awarded to him for the paper "A proof of the Bieberbach conjecture" published in 'Acta Mathematica' in1985. The conjecture itself is simply stated. If

f(z) =z+a_{2}z^{2}+a_{3}z^{3}+a_{4}z^{4}+ ...

converges for|z| < 1and takes distinct values at distinct points of the unit disc, then |a_{n}| ≤ n for all n. Equality is achieved only for the Koebe functionsz/(1+wz)^{2}where w is a constant of absolute value1.

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 the1920s 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. The key was to find norms for which the necessary inequalities could be propagated by the Loewner equation. 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 n to a statement that was almost immediate for very small n, that could be verified numerically for small n, yielding many new cases of the conjecture, and that ultimately revealed itself to be an inequality established several years earlier by Askey and Gasper. The entire construction required a thorough mastery of the literature, formidable analytic imagination, and great tenacity of purpose.

The proof is now available in a form that can be verified by any experienced mathematician as analysis that is "hard" in the original aesthetic sense of Hardy - simple algebraic manipulations linked by difficult inequalities. 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. Without anticipating the future in any way, the Society expresses its appreciation and admiration of past success and wishes him continuing prosperity and good fortune.

At the International Congress of Mathematicians held in Berkeley, California, in August 1986, de Branges was an invited plenary speaker and gave the addressThis report begins with two acknowledgements. One is made to the American Mathematical Society for its continued endorsement of research related to the Bieberbach conjecture. The Steele Prize is only the latest expression of its interest. It should be unnecessary to say that fundamental research cannot be sustained for long periods without the support of learned societies. 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. This mathematical contribution has been obscured by his political allegiance to National Socialism, which caused the mass emigration of German mathematical talent, including many of the great founders of 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. As a research mathematician he exhibited intuitive talent which surpassed his more precise colleagues. Yet the proof of his conjecture is a vindication of their more logical methods.

The proof of the Bieberbach conjecture is difficult to motivate because it is part of a larger research programme whose aim is a proof of the Riemann hypothesis. ...

Difficult problems were left unsolved by the proof of the Bieberbach conjecture. Some of these have since been clarified. Progress has been made, for example, in the structure theory of canonical unitary linear systems and its applications to analytic function theory. Of particular interest is a generalisation of the Beurling inner-outer factorisation. This result is the culmination of a series of publications on canonical unitary linear systems whose state space is a Krein space. They supplement a previous series on canonical unitary linear systems whose state space is a Hilbert space.

Progress has also been made towards the initial objective of a proof of the Riemann hypothesis. The results are conjectured to be also relevant to the proof of the Bieberbach conjecture. A positivity condition has been found for Hilbert spaces of entire functions which is suggested by the theory of the gamma function. The condition appears, for example, in the structure theory of plane measures with respect to which the Newton polynomials form an orthogonal set.

*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. In June 2004 he announced on his website that he had a proof of the Riemann hypothesis and put a 124-page paper up on the website to substantiate the claim. He has continued to revise the paper and also to work on another paper which would prove more general results but have the Riemann hypothesis as a corollary. This paper is now on his website. 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.

The paper [7] gives details of an interview with de Branges. In this interview he gave some fascinating insights into his ways of thinking:-

Let us end this biography by giving two quotes, the first from Atle Selberg [7]:-My mind is not very flexible. I concentrate on one thing and I am incapable of keeping an overall picture. So when I focus on the one thing, I actually forget about the rest of it, and so then I see that at some later time the memory does put it together and there's been an omission. So when that happens then I have to be very careful with myself that I don't fall into some sort of a depression or something like that. You expect that something's going to happen and a major change has taken place, and what you have to realise at that point is that you are vulnerable and that you have to give yourself time to wait until the truth comes out.

Second, let us quote Bela Bollobás who writes [7]:-The thing is it's very dangerous to have a fixed idea. 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. Mathematically he is not the most reliable source in that sense. As I once said to someone - it's a somewhat malicious jest but occasionally I engage in that - after finally they had verified that he had made this result on the Bieberbach Conjecture, I said that Louis de Branges has made all kinds of mistakes, and this time he has made the mistake of being right.

De Branges is undoubtedly an ingenious mathematician, who established his excellent credentials by settling Bieberbach's Conjecture ... Unfortunately, his reputation is somewhat tainted by several claims he made in the past, whose proofs eventually collapsed. I very much hope that this is not the case on this occasion: it is certainly not impossible that this time he has really hit the jackpot by tenaciously pursuing the Hilbert space approach. Mathematics is always considered to be a young man's game, so it would be most interesting if a70-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.

**Article by:** *J J O'Connor* and *E F Robertson*

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