Two-dimensional compact orientable surfaces can be characterized by their genus, which is loosely the number of holes. For example, the surface of a doughnut has one hole, but the surface of a ball has zero holes. One property of a two-dimensional sphere, the surface of a ball, is that any loop on the sphere can be continuously shrunk down to a point without leaving or breaking the surface. This property classifies a collection of certain two-dimensional surfaces.
In 1904, Henri Poincaré asked a corresponding question in three dimensions. This conjecture went unsolved for almost 100 years. In 2000, the Clay Mathematics Institute posted seven unsolved math problems, each with a one million dollar prize attached. These questions are called the millennium problems. One of which is the Poincaré conjecture. An equivalent version of his question is as follows.
If a compact three-dimensional manifold, M, has the property that every simple closed curve can be continuously deformed to a point , is it then true that M is homeomorphic to the 3-sphere?
Here the 3-sphere is the three-dimensional boundary of a four-dimensional ball, such as the set of points , such that .
In 2002 and 2003, Grigori Perelman solved the Poincaré conjecture in a series of articles posted on ArXiv.org. However, Perelman did not seem to care about the prize or the recognition. Astonishingly, he declined the one million dollar prize. In 2006, he was awarded the Fields medal, which is sometimes described as the Nobel prize for mathematics. He rejected the Fields medal as well. Perelman commented, "if the proof was correct then no other recognition was needed."
Perelman may not have desired an award for his work, but he was willing to discuss it. In 2003, he was invite to give a series of talks on his paper at M.I.T., Princeton, and Stony Brook. He accepted all these invitations. When asked about the invitations, he commented simply, "why not?"
Pickover, C. A. (2008), The Math Book. New York, NY: Sterling Publishing Co., Inc.
Nasar, S., Gruber, D. (2006), Manifiold Destiny, The New Yorker, http://www.newyorker.com/magazine/2006/08/28/manifold-destiny