[Paleopsych] Guardian: Maths holy grail could bring disaster for internet
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Maths holy grail could bring disaster for internet
http://www.guardian.co.uk/print/0,3858,5009766-103690,00.html
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
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 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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