Louis de Branges de Bourcia

In 2012, he became a fellow of the American Mathematical Society.

Li 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 It was retracted a few days later, after several mainstream mathematicians exposed a crucial flaw, in a display of interest that his former advisor's claimed proofs have apparently not enjoyed so far.

That original preprint suffered a number of revisions until it was replaced in December 2007 by a much more ambitious claim, which he had been developing for one year in the form of a parallel manuscript. Since that time, he has released evolving versions of two purported generalizations, following independent but complementary approaches, of his original argument. In the shortest of them (43 pages as of 2009), which he titles "Apology for the Proof of the Riemann Hypothesis" (using the word "apology" in the rarely used sense of apologia), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann hypothesis for Dirichlet L-functions (thus proving the generalized Riemann hypothesis) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple. In the other one (57 pages), he claims to modify his earlier approach on the subject by means of spectral theory and harmonic analysis to obtain a proof of the Riemann hypothesis for Hecke L-functions, a group even more general than Dirichlet L-functions (which would imply an even more powerful result if his claim was shown to be correct). As of January 2016, his paper entitled "A proof of the Riemann Hypothesis" is 74 pages long, but does not conclude with a proof. A commentary on his attempt is available on the Internet.

In 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.

Mathematicians remain skeptical, and neither proof has been subjected to a serious analysis. The main objection to his approach comes from a 1998 paper (published two years later) 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 the Riemann hypothesis. 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 the Riemann hypothesis. 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 preprints, but both of them cite a 1986 paper of his that was attacked by Li and Conrey. 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 the Riemann hypothesis is going to yield to functional analysis tools. De Branges, incidentally, also claims that his new proof represents a simplification of the arguments present in the removed paper on the classical Riemann hypothesis, and insists that number theorists will have no trouble checking it. Li and Conrey do not assert that de Branges' mathematics are wrong, only that the conclusions he drew from them in his original papers are, and that his tools are therefore inadequate to address the problems in question.

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.

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 skeptical because de Branges had earlier announced some false results, including a claimed proof of the invariant subspace conjecture in 1964 (incidentally, in December 2008 he 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, 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.

Born to American parents who lived in Paris, de Branges moved to the US 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 PhD in mathematics from Cornell University (1953–57). His advisors were Wolfgang Fuchs and then-future Purdue colleague Harry Pollard. He spent two years (1959–60) at the Institute for Advanced Study and another two (1961–62) at the Courant Institute of Mathematical Sciences. He was appointed to Purdue in 1962.

Louis de Branges de Bourcia (born August 21, 1932) 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's theorem. He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis.