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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 [Poincaré 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] .

Yang–Mills existence and mass gap

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