John Morgan reviews The Poincaré Conjecture by Donald O’Shea, in American Scientist:
A mathematical conjecture is a precise mathematical statement that the formulator believes to be true and important to establish, but for which he or she has no proof. It is posed as a challenge to other mathematicians, and if it is important and difficult, it stimulates much mathematical development before being resolved. One of the most important conjectures in mathematics was formulated in 1904 by the leading mathematician of his day, Henri Poincaré. It was the central, defining problem in the field that he fathered—topology.
For the next 98 years, wave after wave of leading topologists attempted to solve the problem of finding a proof for the Poincaré conjecture. None of them succeeded, but their enormous efforts in this direction were not for naught, because the ideas they developed produced tremendous advances in topology. But through all of it, the Poincaré conjecture remained unapproachable, like the end of the rainbow.
Then in 2002 and 2003, Grigory Perelman, a reclusive Russian mathematician who had disappeared from view in the mid-1990s, posted on the Internet three preprints that claimed to prove the Poincaré conjecture and a great deal more. All the more exciting for mathematicians was the fact that his approach used deep ideas from outside topology. This was no frontal attack, but rather an approach that brought to bear the power of geometry and partial differential equations.