Poincaré Conjecture

The Poincaré Conjecture is a topological problem. It is one of the most well known problems in mathematics, after Fermat's Last Theorem. The problem was formulated by Henri Poincaré in 1904.

Here is the problem, taken from the official problem description (an Adobe Acrobat PDF)

• Consider a compact 3-dimensional manifold V without a boundary.

• Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

The problem is the subject of a one million dollar prize at the Clay Mathematics Institute.

Descriptions of this conjecture are unexpectedly dependent upon international culinary definitions. Most descriptions of the conjecture involve comparing the result of constricting a spherical object with the effects of performing a similar constriction exercise upon a doughnut-shaped object.

Unfortunately, there are two types of doughnut, one of which is a torus or ring (a 'ring doughnut') and the other variety is approximately spherical (called a 'jam doughnut' in the UK because it is typically filled with what the British call jam, and called a jelly doughnut in the USA because they call the same filling jelly) and this doughnut dichotomy has the potential to render the standard explanation of the Poincaré conjecture unintelligibly confusing.

The suspicion is that the two different types of doughnut divide Europe from the United States and this may be in part responsible for the longevity of the unsolved mystery of the problem.

===Recent progress in solving the problem===