What is Poincaré conjecture?
The '''Poincaré Conjecture''' is a question about spheres in [[mathematics]]. It is named after [[Henri Poincaré]], the [[France|French]] mathematician and [[physicist]] who formulated it in 1904. The sphere (also called the 2-sphere, as it is a 2-dimensional [[surface]], although it is usually seen as inside a three dimensional space) has the property that any loop on it can be contracted to a point (if a rubber band is wrapped around the sphere, it is possible to slide it down to a point). Mathematicians say that the 2-sphere is simply connected. Other spaces do not have this property, for example the donut: a rubber band that goes around the whole donut once cannot be slid down to a point without it leaving the surface. Mathematicians knew that this [[property]] was unique to the 2-sphere, in the sense that any other simply connected space that does not have edges and is small enough (in mathematician terms, that is compact) is in fact the 2-sphere. It is no longer true if we remove the idea of smallness however, as an infinitely large plane is also simply connected. Also, a regular disk (a circle and its interior) is simply connected, but it has an edge (the bounding circle). The conjecture asks whether the same is true for the 3-sphere, which is an object living naturally in four dimensions. This question motivated much of modern mathematics, especially in the field of [[topology]]. The question was finally settled in 2002 by [[Grigori Perelman]], a Russian mathematician, with methods from [[geometry]], showing that it is indeed true. He was awarded a Fields Medal and the $1 million Millennium Prize for his work, both of which he declined. The Poincaré conjecture can also be extended to higher dimensions: this is the generalised Poincaré conjecture. Surprisingly, it was easier to prove the fact for higher-dimensional spheres: In 1960, Smale proved it to be true for the 5-sphere, 6-sphere and higher. In 1982, Freedman proved that it was also true for the 4-sphere, for which he was awarded a Fields Medal.