The Poincaré conjecture, a question about the properties of spheres, was one of seven Millennium Prize problems declared by the Clay Mathematics Institute in 2000. In 2003, Grigori Perelman published a proof of the conjecture, which was later confirmed by other mathematicians. The solution has been seen as a major breakthrough in mathematics and was named Science magazine’s scientific breakthrough of the year in 2006.
The Poincaré conjecture is one of the most important conjectures in modern mathematics and has currently been proved adequately to the point of being considered a complete theorem. It is one of seven Millennium Prize problems declared by the Clay Mathematics Institute in 2000. To date, it is the only one of the Millennium Prize problems to have been solved, and its solution has been seen as one of the most important discoveries of the new millennium .
In the early 20th century, a French mathematician, Henri Poincaré, began plotting what would serve as the basis for the mathematical field of topology. One of his main focus was on the properties of spheres, and he spent a lot of attention and energy to delineate the sphere. He asked a series of questions, but the most famous was formulated as: “Consider a compact three-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even if V is not homeomorphic to the three-dimensional sphere?” Although he never made a concrete statement one way or the other, this would come to be known as the Poincaré conjecture.
The most common form of the Poincaré conjecture is simply: every closed simply connected 3-manifold is homeomorphic to the 3-sphere. The Poincaré Conjecture has also been generalized to dimensions higher than three, of the n-sphere form. Although it was originally thought that the Poincare Conjecture itself would be true, it was thought that the generalized Poincare Conjecture would turn out to be false. It therefore came as a surprise when in 1961 the generalized Poincaré conjecture was proved for dimensions greater than four, and then in 1982 when the case of 4 spheres was shown to be true.
In 1982 Richard Hamilton proved 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 prize of $1,000,000 US Dollars (USD) for a satisfactory solution. In 2002 and 2003, mathematician Grigori Perelman published two papers sketching 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 as detailed evidence. Eventually they expanded it into a book on the Poincaré conjecture, and in 2006 Morgan stated that Perelman had solved the problem in 2003. For his work, Perelman was awarded the Fields Medal, but he declined. Even though he has technically settled the Millennium Prize as well, and thus is eligible to receive the million dollars, he has not taken the necessary steps to claim the prize.
Solving the Poincaré Conjecture has been seen as a major breakthrough in mathematics and one of the most important tests of the new millennium. In late 2006, Science magazine named the solution to the Poincaré conjecture as its scientific breakthrough of the year. This was the first time the honor had been bestowed on a breakthrough in pure mathematics.
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