The Poincaré Conjecture is one of the most important conjectures in modern mathematics, and has currently been demonstrated adequately to the point that it is considered a full theorem. It is one of the seven Millennium Prize Problems, stated by the Clay Mathematics Institute in 2000. To date, it is the only of the Millennium Prize Problems to have been solved, and its solution was seen as one of the most important discoveries of the new millennium.
In the early 20th century, a French mathematician, Henri Poincaré, began laying out what would serve as the groundwork for the mathematical field of topology. One of his main focuses was on the properties of spheres, and he spent a great deal of attention and energy on outlining the sphere. He posed a number of questions, but the most famous was phrased as: “Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?” Although he never made a concrete statement one way or another, this would come to be known as the Poincaré Conjecture.
The more common form of the Poincaré Conjecture is simply: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. The Poincaré Conjecture was also generalized to dimensions above three, of the form n-sphere. Although originally it was thought that the Poincaré Conjecture itself would be true, it was thought that the Generalized Poincaré Conjecture would turn out to be false. It was therefore a surprise when the Generalized Poincaré Conjecture was proved for dimensions larger than four in 1961, and then in 1982 when the 4-sphere case was shown to be true.
In 1982 Richard Hamilton showed that the Poincaré Conjecture was true in a number of specialized cases, but was unable to prove it more generally. In 2000 the Clay Mathematics Institute included the Poincaré Conjecture in its Millennium Prize Problems, offering a $1,000,000 US Dollar (USD) prize for a solution proved satisfactorily. In 2002 and 2003 the mathematician Grigori Perelman published two papers that laid out a sketch for a proof of the Poincaré Conjecture.
In 2006 a number of working groups filled in small incidental gaps in Perelman’s work, and John Morgan and Gang Tian wrote it up as a detailed proof. They eventually expanded this into a book about the Poincaré Conjecture, and in 2006 Morgan declared that Perelman had solved the problem in 2003. For his work, Perelman was award the Fields Medal, but he refused it. Although he technically solved the Millennium Prize as well, and therefore is eligible to receive the $1 million USD, he has not undertaken the steps necessary to claim the prize.
The solving of the Poincaré Conjecture was looked at as a great breakthrough in mathematics, and one of the more important proofs of the new millennium. At the end of 2006, the magazine Science named the solution to the Poincaré Conjecture as the Scientific Breakthrough of the Year. This was the first time that the honor had ever been bestowed on a breakthrough in pure mathematics.