[Paleopsych] LRB: Karl Sabbagh: The Strange Case of Louis de Branges
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Karl Sabbagh: The Strange Case of Louis de Branges
http://www.lrb.co.uk/v26/n14/print/sabb01_.html
Two years ago I wrote a book about the Riemann Hypothesis (for an
account of the hypothesis see [14]A.W. Moore's article in this issue).
The proof of it is something that every mathematician would love to
discover or - very much second best - see someone discover. One of the
people I interviewed was Louis de Branges, a Franco-American
mathematician at Purdue University in Lafayette, Indiana, with one
significant proof already under his belt. De Branges was convinced
that a mathematical field in which he was the acknowledged specialist
would lead to a proof of the hypothesis. I have stayed in touch with
him, and earlier this year, he told me he was putting the finishing
touches to a proof he has been working on for 25 years. On 28 April
this proof was published on the internet for other mathematicians to
see and criticise: [15]www.math.purdue.edu/~branges.
There is no evidence that, so far, any mathematician has read it: de
Branges and his proof appear to have been ostracised by the
profession. I have talked to a number of mathematicians about him and
his work over the last few years and it seems that the profession has
come to the view that nothing he does in this area will ever bear
fruit and therefore his work can be safely ignored. It may be that a
possible solution of one of the most important problems in mathematics
is never investigated because no one likes the solution's author.
De Branges's paper was slipped onto the internet without a fuss. Had
he been any other mathematician, there would have been rumours
beforehand. Over the last three years I have got to know a number of
the key men - they are all men - who work on the Riemann Hypothesis,
and although each of them keeps his cards close to his chest, they are
all desperate to get a look at the other fellow's hand. I spoke to
about twenty of the mathematicians most likely to prove the
hypothesis, and if any of them was within reach of a proof, the others
would be agog to see what he was doing. Except, of course, in the case
of de Branges.
De Branges wrote to me in February this year telling me that he was
ready to publish his proof. He doesn't use email, and writes all his
letters by hand.
The final form of the proof of the Riemann Hypothesis will follow
closely the treatment of Hilbert spaces of entire functions which I
discovered in the years 1957-62 and which was published as a book
in 1968. An elegant proof is given which should cause no difficulty
for verification. A reader does, however, need to acquire a broad
knowledge of these spaces to read the argument.
He mentioned several mathematicians he thought would have this broad
knowledge, and went on:
I will give a copy of the manuscript to Paul Malliavin as editor of
the Journal of Functional Analysis. But there is no certainty that
he will consider the paper for publication over the next few
months. Each of them will certainly say that it contains material
relevant to his special interests. They will certainly guard
themselves against any assertion that the argument is correct.
When de Branges told me his proof was complete I suspected that his
paper would be dismissed without being read. Sure enough, in early
May, after the internet publication, when reporters from New Scientist
and Nature started to look into it and to consider whether this really
was the most important mathematical discovery of the last hundred
years, their own mathematical contacts assured them that it could
safely be ignored. But none of these mathematicians claimed to have
actually read de Branges's paper.
The first thing to say about this odd situation is that de Branges is
not a crank. Most mathematicians working on this problem receive a
regular stream of alleged proofs from people with little or no grasp
of number theory or complex analysis, the most likely fields from
which a proof was expected to emerge. Most of these 'proofs' go into
the bin unread. But on the basis of track record, ability and
originality of thought, de Branges is in a very different category.
He may not be a crank, but he is cranky. 'My relationships with my
colleagues are disastrous,' he told me. And he does seem to have left
a trail of disgruntled, irritated and even contemptuous colleagues
behind him if only because he makes no concessions to students and
colleagues who are not familiar with the field in which he works. It
may be a field largely of his own devising, but it makes a genuine
contribution to pure mathematics. When he's fortunate enough to have
students to teach he makes them work their way through a series of
extremely tough exercises and sees no reason to make it easy for them.
He is a person of strict routine: it's the only way he can create the
right conditions for the mathematical thought processes which take up
most of his waking life. Adherence to rules is very important. When I
was walking with him in France, he remonstrated with me because I
stepped on a zebra crossing when two cars were at least a hundred
yards away. 'The cars have to stop if you are on the crossing,' he
said, 'and one of them might have driven into the back of the other.'
He only ever watches one TV programme, the CBS news. 'We cannot afford
the time for more television,' he says.
He is also disarmingly honest. He's even honest about how honest he
is:
I differ from other mathematicians in that I seem to have a deep
honesty that other people sometimes don't have, and it's rather
curious because I certainly didn't have that intention as a young
man. I think I was by nature somebody that would easily cut a
corner - especially if I didn't think it was very important - not
for any real advantage, but I would choose to do so. But the way my
life has evolved, against my own inclinations, I have turned out to
have an unusual probity.
People who keep telling you how modest they are are not usually modest
at all. De Branges isn't like that: he is honest. There have been
occasions when he has told me things that other people would think
twice about revealing. 'My mind is not very flexible,' he once said:
I concentrate on one thing and I am incapable of keeping an overall
picture. So when I focus on the one thing, I actually forget about
the rest of it, and so then I see that at some later time the
memory does put it together and there's been an omission. So when
that happens then I have to be very careful with myself that I
don't fall into some sort of a depression or something like that.
You expect that something's going to happen and a major change has
taken place, and what you have to realise at that point is that you
are vulnerable and that you have to give yourself time to wait
until the truth comes out.
This kind of single-mindedness can be seen in people with Asperger's
Syndrome.
Occasionally he has surprised me by talking about his personal life.
For example, on one occasion he embarked on the story of his first
marriage and ended up telling me how he likes to whistle tunes in the
street. It provides a good example of the rather formal way he speaks:
I'd married a student from Bryn Mawr College, and all of a sudden
she just left, asking for a very substantial amount of money which
I didn't in any way contest. And then staying around in Lafayette
for about ten years, that greatly created a circle of opposition
within the community, because, you see, I was a person that was
seen as being in the wrong by my colleagues, and also by the
community.
The divorce was seen as a criticism of myself, of my performance.
To give you an example of that, my wife sang in an organisation
called the Bach Chorale, and I was seen by musical people as being
somebody that would be against the musical traditions or the arts.
Well, this is a curious thing: I happen to be very musical, I
simply don't have a musical education because of the war years. My
musical qualities are expressed by my whistling. Usually, you know,
when you whistle you disturb people, and I apologise for that, but
people like the things that I whistle. They say: 'Oh, that's a nice
melody, I like that.' It happened when I was going to fetch you at
the station, some young lady said: 'Yes, I like to hear that.' I'm
sure that my musical quality is much greater than that of the girl
who divorced me. I used to sing also in a choir, so I can have a
good voice. My speaking voice is rather flat, but my singing voice
is good.
Whatever personal eccentricities de Branges might have, it's hard to
believe they would be enough to make mathematicians who are desperate
for a proof of the Riemann Hypothesis reject the possibility that he
might now have one.
Yet it has been dismissed as 'probably cobblers'. One reason is that
mathematicians seem to think that de Branges has claimed on several
previous occasions to have proved the Riemann Hypothesis and been in
error. 'He has made something of a tradition, I'm told, of emailing
colleagues every September with a new proof he worked up over the
summer,' another mathematician told me. Successive versions of de
Branges's paper were posted on the internet as his ideas evolved. But
it is unlikely that he has ever emailed any colleagues anything. He is
in contact with very few of them and, in any case, doesn't use email.
De Branges has certainly made errors in the past, but it is difficult
to find a mathematician who hasn't. 'The first case in which I made an
error was in proving the existence of invariant subspaces for
continuous transformations in Hilbert spaces,' he told me. 'This was
something that happened in 1964, and I declared something to be true
which I was not able to substantiate. And the fact that I did that
destroyed my career. My colleagues have never forgiven it.'
Since then, de Branges has on one occasion believed that he had a
finished proof of the Riemann Hypothesis, until an error was pointed
out, and he has also believed himself to be near a proof on several
occasions before himself discovering a mistake. But mathematicians are
surely expected to show a degree of objectivity in assessing their
colleagues' work. Even if de Branges were the error-prone sociopathic
curmudgeon some believe him to be, is that really enough to stop
anyone even considering the possibility of a proof of the Riemann
Hypothesis?
Maybe de Branges just isn't a very good mathematician. But it is
generally agreed that he did solve another important mathematical
problem, the Bieberbach Conjecture, in 1985. Not only that: there are
uncanny similarities between the initial reaction of other
mathematicians to his claim to have proved the Bieberbach Conjecture
then, and the unwillingness now to consider that he might have proved
the Riemann Hypothesis. 'It would be easy to dismiss de Branges as a
crank,' one mathematician wrote on the internet, 'but he has earned
the right to a hearing because the early dismissals of his work on the
Bieberbach Conjecture turned out to be wrong.'
'I am sure that Louis de Branges's many "wrong" proofs of the Riemann
Hypothesis and other conjectures are as chock-full of brilliant ideas
as is his proof of Bieberbach,' another wrote.
A third, in a festschrift to celebrate de Branges's Bieberbach
Conjecture proof, said: 'In March of 1984 the message began to travel.
Louis de Branges was claiming a proof of the Bieberbach Conjecture.
And his method had come from totally unexpected sources: operator
theory and special functions. The story seemed fantastic at the time,
but it turned out to be true.'
'Bieberbach was a tremendous achievement, there's no question about
it,' Peter Sarnak of the Institute for Advanced Studies says. 'Louis
de Branges hit the big time there, really. It was a great problem . .
. and his solution was absolutely brilliant, really brilliant.' But
Sarnak is one of many who dismiss his Riemann Hypothesis proof.
Atle Selberg, one of the greatest pure mathematicians of modern times,
said to me:
The thing is it's very dangerous to have a fixed idea. A person
with a fixed idea will always find some way of convincing himself
in the end that he is right. Louis de Branges has committed a lot
of mistakes in his life. Mathematically he is not the most reliable
source in that sense. As I once said to someone - it's a somewhat
malicious jest but occasionally I engage in that - after finally
they had verified that he had made this result on the Bieberbach
Conjecture, I said that Louis de Branges has made all kinds of
mistakes, and this time he has made the mistake of being right.
De Branges is now claiming to have solved another, far more
significant problem than the Bieberbach Conjecture, again from
'totally unexpected sources', and again most people are treating the
story as fantastic. Will the mathematical community again come to
accept the proof?
It seems unlikely, since there is no one who has read the 121-page
paper all the way through who is competent to judge it. Because de
Branges's proof uses mathematical tools in which he is one of the few
experts, the amount of study required even to become familiar with
those tools before embarking on reading the paper seems too great for
anyone to commit the time. Even the few people who know and understand
de Branges and his method see it as a daunting task. Nikolai Nikolski
helped with the validation of the Bieberbach Conjecture proof, a task
that took a team of mathematicians at the Steklov Institute in
Leningrad several months. 'The Riemann Hypothesis is much more
complicated than the Bieberbach Conjecture,' Nikolski told me.
So you have to be more enthusiastic if you want to validate the
proof. You need to have a team of really enthusiastic high-level
people. De Branges found in the middle of the 1980s the only place
in the world where there were some curious people who just love to
solve complicated problems and who were ready to spend a half a
year on it. He has asked me several times if it's possible to
organise some people to do the same thing with the Riemann
Hypothesis. I love him, so I said to him: 'Yes, if you have a very
huge grant, probably not so huge as in America, to pay, for
instance, the same place in Petersburg.'
But there are plenty of influential mathematicians who just think the
whole process would be a waste of time. Brian Conrey, the director of
the American Institute for Mathematics, who is developing his own
ideas for a proof of the Riemann Hypothesis, is insistent: 'I just
know it can't come out of de Branges's approach,' he said. 'It's the
wrong theory.' But he added a complimentary afterthought: 'If only he
was to market his results for what they are - it is a very beautiful
theory.'
Bela Bollobas, a fellow of Trinity College, Cambridge who teaches at
the University of Memphis is less dogmatic:
De Branges is undoubtedly an ingenious mathematician, who
established his excellent credentials by settling Bieberbach's
Conjecture . . . Unfortunately, his reputation is somewhat tainted
by several claims he made in the past, whose proofs eventually
collapsed. I very much hope that this is not the case on this
occasion: it is certainly not impossible that this time he has
really hit the jackpot by tenaciously pursuing the Hilbert space
approach. Mathematics is always considered to be a young man's
game, so it would be most interesting if a 70-year-old
mathematician were to prove the Riemann Hypothesis, which has been
considered to be the Holy Grail of mathematics for about a hundred
years.
When I visited de Branges in his flat near Paris in May, he did not
behave like a man who was in sight of a million-dollar prize. But this
was not because of any doubts about his proof. 'The proof is there,'
he said, 'but it's just part of a longer paper on the zeta functions.
That's the important work. It's a theory that could lead to a new
understanding of quantum physics, for example, since the way I
approach the subject uses a type of mathematics - spectral theory -
that seems to underlie the behaviour of atoms.'
I asked him how he felt, now that he 'knew' the Riemann Hypothesis was
correct, expecting some expression of satisfaction, or even
exhilaration. 'It's a question of sanity,' he said. 'When you have a
wife who doesn't understand what you do and just wants you out of the
house' (his former wife - he is now happily married); 'when you have a
mother who comes to live with you to look after you and can't
understand what you are doing; when you have colleagues who ignore or
dismiss your work . . .' His voice tapered off. 'I just hope someone
doesn't come along now with an elementary proof of the Riemann
Hypothesis.'
I was puzzled by this. It can't have been a matter of priority, since
his proof is now out on the internet, dated 28 April, and if it is
verified he will get the credit for it. But it turned out he was
worried that were someone else to prove the hypothesis without using
his broader theory of zeta functions, his life's work would be
sidelined as people focused on the other proof and ignored the new
insights he felt he had achieved. 'That would be a disaster,' he said.
Perhaps one day a young mathematician steeped in de Branges's theory
of Hilbert spaces of entire functions will pick up his paper and begin
to work through it. Or perhaps, as the news spreads that the entire
mathematical profession is turning its back on what could be the most
important development in the last hundred years of mathematics, one or
two practitioners will be shamed into reading through de Branges's
proof, just in case he really has cracked an important problem for the
second time in his working life.
[16]Karl Sabbagh is a writer and TV producer whose book The Riemann
Hypothesis is just out in paperback from Farrar, Straus in the US. He
is completing a history of Palestine, to be published by Atlantic.
References
14. http://www.lrb.co.uk/v26/n14/moor02_.html
15. http://www.math.purdue.edu/~branges
16. http://www.lrb.co.uk/contribhome.php?get=sabb01
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