The Poincaré conjecture is a proposition in topology put forward by Henri Poincaré in 1904. Poincaré was led to make his conjecture during his pioneering work in topology, the mathematical study of the properties of objects that stay unchanged when the objects are stretched or bent. In loose terms, the conjecture is that every three-dimensional object that has a set of sphere-like properties (i.e., is topologically equivalent to a sphere) can be stretched or squeezed until it is a three-dimensional sphere (a 3-sphere) without tearing (i.e., making a hole) it. Strictly speaking, the conjecture says that every closed, simply-connected three-manifold is homeomorphic to the three-sphere. It is now known to be true.

Poincaré himself proved the two-dimensional case and he guessed that the principle would hold in three dimensions. Determining if the Poincaré conjecture is correct had been widely judged the most important outstanding problem in topology – so important that, in 2000, the Clay Mathematics Institute in Boston named it as one of seven Millennium Prize Problems and offered a $1 million prize for its solution. Since the 1960s, mathematicians showed by various means that the generalized conjecture is true for all dimensions higher than three – the four-dimensional case finally falling in 1982. But none of these strategies worked in three dimensions. On 7 April 2002 came reports that the Poincaré conjecture might have been proved by Martin Dunwoody of Southampton University, but within a few days a fatal flaw was found in his proof. Then, in April 2003, what appeared to be a genuine breakthrough emerged during a series of lectures delivered at the Massachusetts Institute of Technology by the reclusive Russian mathematician Grigori Perelman of the Steklov Institute of Mathematics (part of the Russian Academy of Sciences in St. Petersburg). His lectures, entitled "Ricci Flow and Geometrization of Three-Manifolds," constituted Perelman's first public discussion of important results contained in two earlier preprints. Mathematicians scrutinized the validity of Perelman's work (which does not actually mention the Poincaré conjecture by name). Finally, it was agreed that Perelman's proof was watertight. However, in 2010 Perelman turned down the $1 million Clay Institute prize – just as he had the prestigious Fields Medal four years earlier.