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In 1904, Poincaré conjectured that any finite-sized 3-dimensional space where you could pull tight all loops must be a 3-sphere.

In 2000, the Clay Mathematics Institute offered a $1,000,000 prize for proving this. The prize may go to Grigori Perelman, a reclusive Russian who seems to have done the job in 2003. But, he didn't bother publishing his papers...










The precise statement of the Poincaré conjecture is that any compact connected 3-dimensional manifold in which all loops are contractible is a 3-sphere.

The background picture is from the Wikipedia article on the 120-cell. It was created by Fritz Obermeyer, who released it into the public domain.