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Grigori Perelman

Until the autumn of 2002, Perelman was best known for his work in [comparison theorem] s in [Riemannian geometry] . Among his notable achievements was a short and elegant proof of the [soul conjecture] .

** The problem **

The Poincaré conjecture, proposed by French mathematician [Henri Poincaré] in 1904, was the most famous open problem in [topology] . Any [loop] on a sphere in three dimensions can be contracted to a point; the Poincaré conjecture surmises that any closed three-dimensional [manifold] where any loop can be contracted to a point, is really just a three-dimensional sphere. The analogous result has been known to be true in higher dimensions for some time, but the case of three-manifolds had turned out to be the hardest of them all. Roughly speaking, this is because in topologically manipulating a three-manifold, there are too few dimensions to move "problematic regions" out of the way without interfering with something else.

In 1999, the [Clay Mathematics Institute] announced the [Millennium Prize Problems] – a $1 million prize for the proof of several conjectures, including the Poincaré conjecture. There was universal agreement that a successful proof would constitute a landmark event in the history of mathematics.

** Perelman's proof **

In November 2002, Perelman posted the first of a series

of [eprints] to the [arXiv] , in which he claimed to have outlined a [proof] of the [geometrization conjecture] , of which the [Poincaré conjecture] is a particular case.

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