Text 489, 144 rader
Skriven 2004-09-07 08:24:38 av Rich Gauszka (1:379/45)
Ärende: Math hypothesis solution = internet disaster?
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From: "Rich Gauszka" <gauszka@hotmail.com>


http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html

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

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 (œ563,000) 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 implications."

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 decade.

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 said.

"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

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