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Mathematics

Conjuring with conjectures

Nature volume 423, pages 125–127 (2003)Cite this article

The Clay Mathematics Institute is offering a million dollars for a solution to the Poincaré conjecture, and Grisha Perelman may have found one. What is the conjecture, and why does it matter?

In 1904, Henri Poincaré was making fundamental advances in topology — the multi-dimensional study of mathematical properties such as 'knotted' or 'connected', which are unchanged by continuous deformations. Buried in his work was an unjustified assumption about three-dimensional spaces. When he noticed it, and failed to find a proof, the assumption became first a question, and then a conjecture. Unsolved almost a century later, the Poincaré conjecture joined other problems (such as the Riemann hypothesis and Yang–Mills theory) on a list of 'Millennium Prize Problems' for whose solutions a substantial cash prize is offered1. Grisha Perelman of the Steklov Institute of Mathematics in St Petersburg, Russia, may now have found the answer2,3.

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Figure 1: Testing topology.

References

  1. http://www.claymath.org/Millennium_Prize_Problems

  2. Perelman, G. Preprint math.DG/0211159 at <http://arXiv.org> (2002).

  3. Perelman, G. Preprint math.DG/0303109 at <http://arXiv.org> (2003).

  4. Stewart, I. Flatterland (Macmillan, London, 2001).

    MATH  Google Scholar 

  5. Hamilton, R. S. J. Diff. Geom. 17, 255–306 (1982).

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  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Ian Stewart

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Correspondence to Ian Stewart.

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Stewart, I. Conjuring with conjectures. Nature 423, 125–127 (2003). https://doi.org/10.1038/423124a

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