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Louis de Branges de Bourcia

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 generalized Riemann hypothesis (GRH).

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 (1949–53), and received a Ph.D. in mathematics from Cornell University (1953–7). 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.

An analyst, de Branges has made incursions into real, functional, complex, harmonic (Fourier) and Diophantine analyses. As far as particular techniques and approaches are concerned, he is an expert in spectral and operator theories.

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.

In June 2004, de Branges announced he had a proof of the Riemann hypothesis (RH; 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. The main objection to his approach comes from a 2000 paper authored by 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. Specifically, the authors proved that the positivity required of an analytic function F(z) which de Branges would use to construct his proof would also force it to assume certain inequalities that, according to them, the functions actually relevant to a proof do not satisfy. As their paper predates the current purported proof by five years, and refers to work published in peer-reviewed journals by de Branges between 1986 and 1994, it remains to be seen whether de Branges has managed to circumvent their objections. He does not cite their paper in his preprint. Journalist Karl Sabbagh, who in 2003 had written a book on the Riemann Hypothesis centered on de Branges, quoted Conrey as saying in 2005 that he still believed de Branges' approach was inadequate to tackling the conjecture, even though he acknowledged that it is a beautiful theory in many other ways. He gave no indication he had actually read the then current version of the purported proof (see reference 1). In a 2003 technical comment, Conrey states he does not believe RH is going to yield to functional analysis tools.

Somewhat ironically, Li himself released a purported proof of the Riemann Hypothesis in the arXiv in July 2008. It was retracted a few days later, after several mainstream mathematicians exposed a crucial flaw, in a display of interest that his former advisor's claimed proof has apparently not enjoyed so far.

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.

In December 2007, de Branges withdrew his preprint on the Riemann Hypothesis, effectively replacing it by a much more ambitious claim. For one year he had been releasing evolving versions of a purported generalization of his argument. According to this new preprint, his previous result can be expanded into a proof of RH for Hecke L-functions. These include Dirichlet L-functions as a subgroup, which means that a positive result for them would in effect prove in the affirmative the Generalized Riemann Hypothesis. In that preprint, which as of 2008 has reached 53 pages, he claims to show how to use the arguments from a previous paper on Hilbert spaces to prove GRH; precisely the same paper Li and Conrey had claimed to represent a failed attack on RH. He also claims that his new proof represents a simplification of the arguments present in the removed paper on the classical RH, and that number theorists will have no trouble checking it.

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). This poses another difficulty to verification of his current work, which is largely self-contained: most research papers de Branges chose to cite in his purported proof of RH were written by himself over a period of forty years. During most of his working life, he published articles as the sole author.

It must be noted that the Riemann Hypothesis, although not so popular among pseudomathematicians (it is not easily formulated), is one of the deepest problems in the entire mathematics.It ranks among one of the seven Millennium Prize Problems. A simple search in the arXiv will yield several claims of proofs, some of them by mathematicians working at academic institutions, that remain unverified and are usually dismissed by mainstream mathematicians. A few of those have even cited de Branges' preprints in their references, which means that his work has not gone completely unnoticed. Still, de Branges' case is not unique; but he is probably the most renowned professional to have a current claim.

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.

He has released another preprint in his site that claims to solve a measure problem due to Stefan Banach.

In 1989 he was the first recipient of the Ostrowski Prize and in 1994 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research.

References

External links

See also

Scattering theory - used by de Branges in his early approach to RH. Peter Lax

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