updated 8/22/2006 1:16:14 PM ET 2006-08-22T17:16:14

The Poincare conjecture involves topology, a branch of math that studies shapes. It essentially says that in three dimensions you cannot transform a doughnut shape into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere.

There is a catch: the space has to be finite. Imagine an ant crawling on an apple in a straight line. It can only walk so far before it's back where it started.

That surface is two-dimensional. In three dimensions, shapes are harder to determine because people cannot directly 'see' them and there are many more possible types of holes.

The conjecture is named for French mathematician and physicist Henri Poincare, who proposed it in 1904.

An analogous conjecture was proved for spaces of more than three dimensions over 20 years ago. But the specific 3-D case flummoxed mathematicians for years.

In 1982, Columbia University's Richard Hamilton developed a technique called Ricci flow that mathematically ironed out wrinkles in 3-D surfaces and provided a blueprint for cracking the Poincare conundrum.

A problem was posed by puzzling, dense spots called singularities, which exhibited sudden, uncontrolled change.

Grigory Perelman's breakthrough was to understand how to analyze these singularities, essentially neutralizing them for a while and allowing the Ricci flow to proceed smoothly and show what a given space is really like, topologically speaking.

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