[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