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Millennium Prize Problems

In [topology] , a [sphere] with a two-dimensional [surface] is essentially [characterized] by the fact that it is [simply connected] . It is also true that every 2-dimensional surface which is both [compact] and simply connected is topologically a sphere. The [PoincarĂ© conjecture] is that this is also true for spheres with three-dimensional surfaces. The question had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying [3-manifold] s.

The official statement of the problem was given by [John Milnor] .

A proof of this conjecture was given by [Grigori Perelman] in 2003; its review was completed in August 2006, and Perelman was selected to receive the [Fields Medal] for his solution. Perelman declined that award. Perelman was officially awarded the Millennium prize on March 18, 2010. According to the Daily Mail, Perelman turned down the prize.

** The Riemann hypothesis **

The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the [Riemann zeta function] have a real part of 1/2. A proof or disproof of this would have far-reaching implications in [number theory] , especially for the distribution of [prime number] s. This was [Hilbert's eighth problem] , and is still considered an important open problem a century later.

The official statement of the problem was given by [Enrico Bombieri] .

** Yangâ€“Mills existence and mass gap **

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