Text B. Poincare conjecture (2,000 characters).

In mathematics, the Poincare conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.

Originally conjectured by Henri Poincare, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincare conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof followed on from the program of Richard Hamilton to use the Ricci flow to attempt to solve the problem. Perelman introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way. Several teams of mathematicians have verified that Perelman's proof is correct.

The Poincare conjecture, before being proven, was one of the most important open questions in topology. It is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Perelman was awarded the Millennium Prize on March 18, 2010. On July 1, 2010, he turned down the prize saying that he believes his contribution in proving the Poincare conjecture was no greater than that of Hamilton's (who first suggested using the Ricci flow for the solution). The Poincare conjecture is the only solved Millennium problem.

In 2006, the journal Science honored Perelman's proof of the Poincare conjecture as the scientific "Breakthrough of the Year", the first time this had been bestowed in the area of mathematics.

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Text A. Russian mathematician receives the 2014 Abel Prize (1,800 characters).	\|	Text C. Wavelet theory (2, 500 characters).

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