### De Branges’s Proof of the Bieberbach Conjecture

It is a simple matter to post a paper on arXiv.org claiming to prove Goldbach’s Conjecture, the Twin Primes Conjecture or any of a large number of other interesting hypotheses that are still open. However, unless the person posting the article is well known, it is likely to be completely ignored.

Mathematicians establish their claims and convince their colleagues by submitting their work to peer-reviewed journals. The work is then critically scrutinized and evaluated by mathematicians familiar with the relevant field, and is either accepted for publication, sent back for correction or revision or flatly rejected.

In his book The Proof is in the Pudding: the Changing Nature of Mathematical Proof, Professor Stephen Krantz gives an account of how the concept of proof, which emerged in ancient Greece, has evolved over the past 2,500 years. Today there are clearly-defined procedures for moving from a claim to have a proof to the widespread acknowledgement of its validity.

Krantz’s book is enriched by many historical and modern examples. In this article, we take a look at the unusual course — from proposal to acceptance — of Louis de Branges’s proof of the Bieberbach Conjecture.

The Bieberbach Conjecture

In 1916, Ludwig Bieberbach (described on Wikipedia as “a German mathematician and Nazi”) announced a conjecture about a condition that must hold for any holomorphic function mapping the open unit disk injectively to the complex plane. The Bieberbach Conjecture is a statement about the Taylor coefficients of a one-to-one holomorphic function from ${\{z \in\mathbb{C} : |z| < 1\}}$ into the complex plane. If the function is defined to have a series expansion of the form

$\displaystyle f(z) = z + \sum _{n\geq 2} a_{n}z^{n} \,,$

the conjecture imposes strict limits on the sizes of the coefficients:

$\displaystyle |a_{n}| \leq n \qquad {\text{for all }} n \geq 2 \,.$

This was proven in 1984 by Louis de Branges, and thus Bieberbach’s Conjecture became de Branges’s Theorem.

Louis de Branges, Purdue University.

Louis de Branges (born 1932), a French-American mathematician now at Purdue University, had previously announced other mathematical advances, which turned out to be invalid, so his claim was not immediately believed. However, a team of mathematicians at the Steklov Mathematics Institute in St. Petersburg, where de Branges was on sabbatical leave, forensically analysed his proof and, having filled in a few gaps, accepted the validity of de Branges’s claim. The results were published in 1985 in a 16-page paper in Acta Mathematica. There is no longer any doubt about the correctness of de Branges’s proof.

Upset at the Conference

In The Proof is in the Pudding, Krantz tells the intriguing story of how things evolved. Before his article appeared in the journal, de Branges distributed a large number of preprints to colleagues. Two recipients, Christian Pommerenke and Carl Fitzgerald, examined the proof and found several significant simplifications and clarifications. These were substantial enough to justify a separate paper.

Unfortunately, the paper of Fitzgerald and Pommerenke appeared in print before de Branges’s paper was published. It is noteworthy that the title of this paper was “The de Branges theorem on univalent functions” so, clearly, there was no intention to deny the honour of first proof to de Branges. A special conference was convened to celebrate the work of de Branges, but, when Carl Fitzgerald introduced de Branges to address the conference, things went awry: “Professor de Branges stood up and announced to one and all that Fitzgerald and his collaborators were gangsters who were set to steal his ideas” (Krantz, 2011, pg 165).

Surprising Developments

Around 1990, a student at Stanford, Lenard Weinstein, produced, in his thesis, a four-page proof of the Bieberbach Conjecture, using no more than straightforward arguments of calculus. And in 1993, Doron Zeilberger and Shalosh B. Ekhad produced a computer-assisted proof of the Bierberbach Conjecture that was just a few lines long. As Krantz informs us, Shalosh B. Ekhad is not a member of Homo Sapiens, but is Zeilberger’s name for his computer! Further fascinating details of this sorry affair are provided in Krantz (2011).

Louis de Branges has claimed to have proved the Riemann Hypothesis. He has produced documents in support of this claim: an 89-page paper (Filename: proof-riemann-2017-04.pdf) on RH is posted on his website at Purdue. As of now, the consensus view is that the hypothesis remains open.

Sources

${\bullet}$ Louis de Branges’s publications at Purdue.

${\bullet}$ Krantz, Stephen G., 2011: The Proof is in the Pudding: the Changing Nature of Mathematical Proof. Springer, 264pp. ISBN: 978-0-3874-8908-7

${\bullet}$ Wikipedia article: De Branges’s Theorem.