Louis de Branges de Bourcia
Encyclopedia
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' 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.
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, 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, 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 space
s of entire function
s, 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.
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 generalisations, 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-function
s (thus proving GRH) 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 RH for Hecke L-functions, a group even more general than Dirichlet L-functions (which would imply an even more powerful result if his claim were proved correct).
Mathematicians remain sceptical, 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 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 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 RH 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 RH, and insists that number theorists will have no trouble checking it. It must be stressed that 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.
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 proofs have apparently not enjoyed so far.
Meanwhile, the Apology has 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 proofs. 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.
The particular analysis tools he has developed, although largely successful in tackling the Bieberbach conjecture, have 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 six unsolved 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 scholars. A few of those have even cited de Branges' preprints in their references, which means that his work has not gone completely unnoticed. This shows that de Branges' apparent estrangement is not an isolated case, but he is probably the most renowned professional to have a current unverified 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.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He is the Edward C. Elliott Distinguished Professor of Mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at Purdue University
Purdue University
Purdue University, located in West Lafayette, Indiana, U.S., is the flagship university of the six-campus Purdue University system. Purdue was founded on May 6, 1869, as a land-grant university when the Indiana General Assembly, taking advantage of the Morrill Act, accepted a donation of land and...
in West Lafayette
West Lafayette, Indiana
As of the census of 2010, there were 29,596 people, 12,591 households, and 3,588 families residing in the city. The population density was 5,381.1 people per square mile . The racial makeup of the city was 74.3% White, 17.3% Asian, 2.7% African American, 0.16% Native American, 0.03% Pacific...
, Indiana
Indiana
Indiana is a US state, admitted to the United States as the 19th on December 11, 1816. It is located in the Midwestern United States and Great Lakes Region. With 6,483,802 residents, the state is ranked 15th in population and 16th in population density. Indiana is ranked 38th in land area and is...
. He is best known for proving the long-standing Bieberbach conjecture
De Branges' theorem
In complex analysis, the Bieberbach conjecture or de Branges's theorem, posed by and proven by , states a necessary condition on a holomorphic function to map the open unit disk of the complex plane injectively to the complex plane....
in 1984, now called de Branges' theorem. He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis
Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
(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
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
. He did his undergraduate studies at the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
(1949–53), and received a Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
in mathematics from Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
(1953–7). His advisors were Harry Pollard and Wolfgang Fuchs
Wolfgang Heinrich Johannes Fuchs
Wolfgang Heinrich Johannes Fuchs was a mathematician specializing in complex analysis. His main area of research was Nevanlinna theory....
. He spent two years (1959–60) at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
and another two (1961–2) at the Courant Institute of Mathematical Sciences
Courant Institute of Mathematical Sciences
The Courant Institute of Mathematical Sciences is an independent division of New York University under the Faculty of Arts & Science that serves as a center for research and advanced training in computer science and mathematics...
. He was appointed to Purdue in 1962.
An analyst, de Branges has made incursions into real
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
, functional
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, complex
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, harmonic
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
(Fourier) and Diophantine analyses. As far as particular techniques and approaches are concerned, he is an expert in spectral
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
and operator
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
theories.
Work
De Branges' proofMathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
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, 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
Steklov Institute of Mathematics
Steklov Institute of Mathematics or Steklov Mathematical Institute is a research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. It was established April 24, 1934 by the decision of the General Assembly of the Academy of Sciences of the USSR in...
in Leningrad
Leningrad
Leningrad is the former name of Saint Petersburg, Russia.Leningrad may also refer to:- Places :* Leningrad Oblast, a federal subject of Russia, around Saint Petersburg* Leningrad, Tajikistan, capital of Muminobod district in Khatlon Province...
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 space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s of entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
s, 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
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
(RH; often called the greatest unsolved problem in mathematics) and published the 124-page proof on his website.
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 generalisations, 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
Apologetics
Apologetics is the discipline of defending a position through the systematic use of reason. Early Christian writers Apologetics (from Greek ἀπολογία, "speaking in defense") is the discipline of defending a position (often religious) through the systematic use of reason. Early Christian writers...
), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s (thus proving GRH) 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 RH for Hecke L-functions, a group even more general than Dirichlet L-functions (which would imply an even more powerful result if his claim were proved correct).
Mathematicians remain sceptical, 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
Brian Conrey
John Brian Conrey is an American mathematician and the executive director of the American Institute of Mathematics. His research interests are in number theory, specifically analysis of L-functions and the Riemann zeta function. He received his B.S. from Santa Clara University in 1973 and received...
and Xian-Jin Li, one of de Branges' former Ph.D. students and discoverer of Li's criterion
Li's criterion
In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is completely equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li, who presented it in 1997. Recently, Enrico Bombieri and Jeffrey C...
, a notable equivalent statement of RH. Peter Sarnak
Peter Sarnak
Peter Clive Sarnak is a South African-born mathematician. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics...
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 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 RH 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 RH, and insists that number theorists will have no trouble checking it. It must be stressed that 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.
Somewhat ironically, Li himself released a purported proof of the Riemann Hypothesis in the arXiv
ArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...
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.
Meanwhile, the Apology has 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 proofs. 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.
The particular analysis tools he has developed, although largely successful in tackling the Bieberbach conjecture, have 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 six unsolved Millennium Prize Problems
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
. A simple search in the arXiv
ArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...
will yield several claims of proofs, some of them by mathematicians working at academic institutions, that remain unverified and are usually dismissed by mainstream scholars. A few of those have even cited de Branges' preprints in their references, which means that his work has not gone completely unnoticed. This shows that de Branges' apparent estrangement is not an isolated case, but he is probably the most renowned professional to have a current unverified 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
De Branges space
In mathematics, a de Branges space is a concept in functional analysis and is constructed from a de Branges function....
.
He has released another preprint in his site that claims to solve a measure problem due to Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....
.
In 1989 he was the first recipient of the Ostrowski Prize
Ostrowski Prize
The Ostrowski Prize is a mathematics award given every other year by an international jury from the universities of Basel, Jerusalem, Waterloo and the academies of Denmark and the Netherlands...
and in 1994 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research.
External links
- Louis de Branges at the Mathematics Genealogy ProjectMathematics Genealogy ProjectThe Mathematics Genealogy Project is a web-based database for the academic genealogy of mathematicians. As of September, 2010, it contained information on approximately 145,000 mathematical scientists who contribute to "research-level mathematics"...
- Papers by de Branges, including all his purported proofs (personal homepage, includes list of peer-reviewed publications).