The Poincaré conjecture (pdf) is one of seven Millennium Problems proposed by CMI in 2000 to highlight research on some of the most difficult problems facing mathematicians at the turn of the second millennium. It is the first problem to be certified as solved.
"Resolution of the Poincaré conjecture by Grigoriy Perelman brings to a close the century-long quest for the solution," CMI President James Carlson said. "It is a major advance in the history of mathematics that will long be remembered."
Formulated in 1904 by Henri Poincaré, the conjecture concerns three-dimensional shapes (compact manifolds) contained in four-dimensional space. Because such objects cannot be visualized directly, Poincaré asked whether there is a test for recognizing when a shape is a three-dimensional sphere (or three-sphere), no matter how distorted that shape may be.
Perelman succeeded in proving the conjecture, taking advantage of a number of relatively recent advances, including use of the Ricci flow method pioneered and developed by Richard Hamilton. In 2006, Perelman received the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow."
CMI and the Institut Henri Poincaré will hold a conference on June 8-9, 2010, in Paris to celebrate the Poincaré conjecture and its resolution.
Full press release from CMI (March 18, 2010) available here (pdf).