Henri Poincare (1854-1912) was one of the most eminent French mathematicians of the past two centuries.
One of Poincare’s best-known problems is what is today called the Poincare conjecture.
The poincare conjecture is considered so important that the clay Mathematics Institute named it one of the seven millennium prize problems will be awarded $1 million.
The Poincare conjecture falls within the realm of topology.
This branch of mathematics focuses, roughly speaking, on the issue of whether one body can be deformed into a different body through pulling, squashing or rotating, without tearing or gluing pieces together.
A ball, an egg, and a flowerpot are, topologically speaking, equivalent bodies, since any of them can be deformed into any of others without performing any of the “illegal” actions.
A ball and a coffee cup, on the other hand, are not equivalent, since the cup has a handle, which could not have been formed out of the ball without poking a hole through it.
The ball, egg, and a flowerpot are said to be “simply connected” as opposed to the cup, a bagel, or a pretzel.
Poincare sought to investigate such issues not by geometric means but through algebra, thus becoming the originator of “algebraic topology.”
In 1904 he asked whether all bodies that do not have a handle are equivalent to spheres. In two dimensions this questions this question refers to the surfaces of eggs, coffee cups, and flowerpots and can be answered yes. (Surfaces like the leather skin of a football or the crust of a bagel are two-dimensional objects floating in three-dimensional space)
For three-dimensional surfaces in four-dimensional space the answer is not quite clear. While Poincare was inclined to believe that the answer was yes, he was not able to provide a proof.
Several Mathematicians were able to prove the equivalent of Poincare’s conjecture for all bodies of dimension greater than four. This is because higher-dimensional spaces provide more elbowroom so mathematicians find it simpler to prove the Poincare conjecture.
But, for three-dimensional surfaces in four-dimensional space –remember: The surface of a four-dimensional object is a three-dimensional object. Poincare’s conjecture remained as elusive as ever.
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