Maths holy grail could bring disaster for internet

R. A. Hettinga rah at
Mon Sep 6 22:52:39 EDT 2004


The Guardian

Maths holy grail could bring disaster for internet

Two of the seven million dollar challenges that have baffled for more than
a century may be close to being solved
Tim Radford, science editor
Tuesday September 7, 2004

The Guardian
Mathematicians could be on the verge of solving two separate million dollar
problems. If they are right - still a big if - and somebody really has
cracked the so-called Riemann hypothesis, financial disaster might follow.
Suddenly all cryptic codes could be breakable. No internet transaction
would be safe.

 On the other hand, if somebody has already sorted out the so-called
Poincaré conjecture, then scientists will understand something profound
about the nature of spacetime, experts told the British Association science
festival in Exeter yesterday.

 Both problems have stood for a century or more. Each is almost dizzyingly
arcane: the problems themselves are beyond simple explanation, and the
candidate answers published on the internet are so intractable that they
could baffle the biggest brains in the business for many months.

 They are two of the seven "millennium problems" and four years ago the
Clay Mathematics Institute in the US offered $1m (£563,000) to anyone who
could solve even one of these seven. The hypothesis formulated by Georg
Friedrich Bernhard Riemann in 1859, according to Marcus du Sautoy of Oxford
University, is the holy grail of mathematics. "Most mathematicians would
trade their soul with Mephistopheles for a proof," he said.

 The Riemann hypothesis would explain the apparently random pattern of
prime numbers - numbers such as 3, 17 and 31, for instance, are all prime
numbers: they are divisible only by themselves and one. Prime numbers are
the atoms of arithmetic. They are also the key to internet cryptography: in
effect they keep banks safe and credit cards secure.

 This year Louis de Branges, a French-born mathematician now at Purdue
University in the US, claimed a proof of the Riemann hypothesis. So far,
his colleagues are not convinced. They were not convinced, years ago, when
de Branges produced an answer to another famous mathematical challenge, but
in time they accepted his reasoning. This time, the mathematical community
remains even more sceptical.

 "The proof he has announced is rather incomprehensible. Now mathematicians
are less sure that the million has been won," Prof du Sautoy said.

 "The whole of e-commerce depends on prime numbers. I have described the
primes as atoms: what mathematicians are missing is a kind of mathematical
prime spectrometer. Chemists have a machine that, if you give it a
molecule, will tell you the atoms that it is built from. Mathematicians
haven't invented a mathematical version of this. That is what we are after.
If the Riemann hypothesis is true, it won't produce a prime number
spectrometer. But the proof should give us more understanding of how the
primes work, and therefore the proof might be translated into something
that might produce this prime spectrometer. If it does, it will bring the
whole of e-commerce to its knees, overnight. So there are very big

 The Poincaré conjecture depends on the almost mind-numbing problem of
understanding the shapes of spaces: mathematicians call it topology.
Bernhard Riemann and other 19th century scholars wrapped up the
mathematical problems of two-dimensional surfaces of three dimensional
objects - the leather around a football, for instance, or the distortions
of a rubber sheet. But Henri Poincaré raised the awkward question of
objects with three dimensions, existing in the fourth dimension of time. He
had already done groundbreaking work in optics, thermodynamics, celestial
mechanics, quantum theory and even special relativity and he almost
anticipated Einstein. And then in 1904 he asked the most fundamental
question of all: what is the shape of the space in which we live? It turned
out to be possible to prove the Poincaré conjecture in unimaginable worlds,
where objects have four or five or more dimensions, but not with three.

 "The one case that is really of interest because it connects with physics,
is the one case where the Poincaré conjecture hasn't been solved," said
Keith Devlin, of Stanford University in California.

 In 2002 a Russian mathematician called Grigori Perelman posted the first
of a series of internet papers. He had worked in the US, and was known to
American mathematicians before he returned to St Petersburg. His proof - he
called it only a sketch of a proof - was very similar in some ways to that
of Fermat's last theorem, cracked by the Briton Andrew Wiles in the last

 Like Wiles, Perelman is claiming to have proved a much more complicated
general problem and in the course of it may have solved a special one that
has tantalised mathematicians for a century. But his papers made not a
single reference to Poincaré or his conjecture. Even so, mathematicians the
world over understood that he tackled the essential challenge. Once again
the jury is still out: this time, however, his fellow mathematicians
believe he may be onto something big.

 The plus: the multidimensional topology of space in three dimensions will
seem simple at last and a million dollar reward will be there for the
asking. The minus: the solver does not claim to have found a solution, he
doesn't want the reward, and he certainly doesn't want to talk to the media.

 "There is good reason to think the kind of approach Perelman is taking is
correct. However there are some problems. He is very reclusive, won't talk
to the press, has shown no indication of publishing this as a paper, which
you would have to do if you wanted to get the prize from the Clay
Institute, and has shown no interest in the prize whatsoever," Dr Devlin

 "Has it been proved? We don't know. We have good reason to assume it has
been and within the next 12 months, in the inner core of experts in
differential geometry, which is the field we are speaking about, people
will start to say, yes, OK, this looks right. But there is not going to be
a golden moment."

 The implications of a proof of the Poincaré conjecture would be enormous,
but like the problem itself, very difficult to explain, he said. "It can't
fail to have huge ramifications: not only the result, but the methods as
well. At that level of abstraction, that level of connection, so much can
follow. Differential geometry is the subject that is really underneath
understanding everything about space and spacetime."

 Seven baffling pillars of wisdom

1 Birch and Swinnerton-Dyer conjecture Euclid geometry for the 21st
century, involving things called abelian points and zeta functions and both
finite and infinite answers to algebraic equations

2 Poincaré conjecture The surface of an apple is simply connected. But the
surface of a doughnut is not. How do you start from the idea of simple
connectivity and then characterise space in three dimensions?

 3 Navier-Stokes equation The answers to wave and breeze turbulence lie
somewhere in the solutions to these equations

 4 P vs NP problem Some problems are just too big: you can quickly check if
an answer is right, but it might take the lifetime of a universe to solve
it from scratch. Can you prove which questions are truly hard, which not?

 5 Riemann hypothesis Involving zeta functions, and an assertion that all
"interesting" solutions to an equation lie on a straight line. It seems to
be true for the first 1,500 million solutions, but does that mean it is
true for them all?

 6 Hodge conjecture At the frontier of algebra and geometry, involving the
technical problems of building shapes by "gluing" geometric blocks together

 7 Yang-Mills and Mass gap A problem that involves quantum mechanics and
elementary particles. Physicists know it, computers have simulated it but
nobody has found a theory to explain it

R. A. Hettinga <mailto: rah at>
The Internet Bearer Underwriting Corporation <>
44 Farquhar Street, Boston, MA 02131 USA
"... however it may deserve respect for its usefulness and antiquity,
[predicting the end of the world] has not been found agreeable to
experience." -- Edward Gibbon, 'Decline and Fall of the Roman Empire'

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