Louis de Branges de Bourcia

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Louis de Branges de Bourcia (born August 21, 1932 in Paris, France) is a French-American mathematician. He is the Edward C. Elliott Distinguished Professor of Mathematics at Purdue University in West Lafayette, Indiana. He is best known for proving the long-standing Bieberbach conjecture in 1984, now called de Branges' theorem. He claims to have proved several important conjectures in mathematics, including the Riemann Hypothesis.

Born to American parents who lived in Paris, de Branges moved to the U.S. in 1941 with his mother and sisters. His native language is French. He did his undergraduate studies at the Massachusetts Institute of Technology, and received a Ph.D. in mathematics from Cornell University in 1957. His advisors were Harry Pollard and Wolfgang Fuchs. He spent two years (1959-60) at the Institute for Advanced Study and another two (1961-2) at the Courant Institute of Mathematical Sciences. He was appointed to Purdue in 1962.

His field is Analysis; more specifically, functional, complex, harmonic and Diophantine analyses. As far as particular techniques and approaches are concerned, he is an expert in spectral and operator theories.

[edit] Work

De Branges' proof of the Bieberbach conjecture was not initially accepted by the mathematical community. Rumors of his proof began to circulate in March 1984, but many mathematicians were sceptical, because de Branges had earlier announced some false results, including a claimed proof of the invariant subspace conjecture in 1964 (incidentally, he recently published a new claimed proof for this conjecture on his website). It took verification by a team of mathematicians at Steklov Institute of Mathematics in Leningrad to validate de Branges' proof, in a process that took several months and led later to significant simplification of the main argument. The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges.

Actually, the correctness of the Bieberbach conjecture was only the most important consequence of de Branges' proof, which covers a more general problem, the Milin conjecture.

On June 2004, de Branges announced he had a proof of the Riemann hypothesis (often called the greatest unsolved problem in mathematics) and published the 124-page proof on his website. He also published an "Apology for the proof of the Riemann hypothesis", which contains a broad explanation of the tools used in the proof.

On December 2005, he reduced his claimed proof to 41 pages. Mathematicians remain sceptical, and the proof has not been subjected to a serious analysis.^[1] The main objection to his approach comes from a 1998 paper authored by John Brian Conrey and Xian-Jin Li, one of de Branges' former Ph.D. students and discoverer of Li's criterion, a notable equivalent statement of RH. Peter Sarnak also gave contributions to the central argument. The paper - which, contrarily to de Branges' claimed proof, was peer-reviewed and published in a scientific journal - gives numerical counterexamples and non-numerical counterclaims to some positivity conditions concerning Hilbert spaces which would, according to previous demonstrations by de Branges, imply the correctness of RH. However, neither of the authors went so far as to state in the paper that it would be impossible to address the problem this way; they merely presented strong obstacles and corrected some of de Branges' assumptions. Their paper predates the current purported proof by five years.

The Apology has since become a diary of sorts, in which he also discusses the historical context of the Riemann Hypothesis, and how his personal story is intertwined with the proof. He signs his papers and preprints as "Louis de Branges", and is always cited this way. However, he does seem interested in his de Bourcia ancestors, and discusses the origins of both families in the Apology.

Other proofs de Branges claims to have found are for a measure problem due to Stefan Banach - thus, an incursion into measure theory - and an ongoing work on equivalent statements of RH for Hecke L-functions, which he claims will simplify the understanding of his own proof of the classical Riemann Hypothesis when completed. Hecke functions include Dirichlet L-functions as a subgroup, and a positive result of RH for the former would thus amplify the impact of de Branges' work by solving in the affirmative the much more important Generalized Riemann Hypothesis.

He is a lonely investigator even by mathematical standards. Of his eighty or so scientific papers, as listed in MathSciNet, he had only three collaborators in a total of six articles. In a career spanning half a century, he has had less than a dozen doctoral students. The same degree of isolation applies to his techniques. The particular approach he has developed to functional analysis, although largely successful in tackling the Bieberbach conjecture, has been mastered by only a handful of other mathematicians (many of whom have studied under de Branges), which poses another difficulty to verification of his current work. His work is largely self-contained: all research papers de Branges chose to cite in his purported proof of RH were written by himself.

Two named concepts arose out of de Branges' work. An entire function satisfying a particular inequality is called a de Branges function. Given a de Branges function, the set of all entire functions satisfying a particular relationship to that function, is called a de Branges space.

In 1989 he was the first recipient of the Ostrowski Prize, and used the award to visit a research institute in Germany and conduct calculations on the Riemann zeta function alongside professor Eberhard Freitag. In 1994 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research.