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Waiting for the Poincare proof.

Waiting for the Poincare proof

To mathematicians, especially topologists, proving the Poincare conjecture would be something like being the first to climb Mt. Everest. For more than 80 years, numerous mathematicians have stumbled over this infamous problem, always slipping somewhere along the way. Sometimes, only a tiny gap -- a subtel error buried within pages of mathematics -- has halted the ascent.

The latest claim, reported in the March 20 NATURE, comes from the University of Warwick in Coventry, England. There, Colin Rourke and Portuguese graduate student Eduardo Rego recently announced success in proving the Poincare conjecture. However, no one else has yet vertified the proffered proof.

"There's some skepticism in the community because there have been many false proofs," says mathematician Joan S. birman of Columbia University in New York City. "So people are weighing it carefully before deciding whether this is a proof or not. Nevertheless, it seems to have passed some tests."

Simply put, the Poincare conjecture proposes that no matter how distorted or twisted its shape may be, any object that mathematically behaves like a three-dimensional sphere is a three-dimensional sphere. Although this sounds obvious, the difficulty lies in the enumeration of all the different ways in which three-dimensional space can be stretched and molded to form geometric objects.

Over the years, topologists have invented a variety of techniques, including "surgery" and "handle theory," to classfy and characterize all these objects and to show which shapes are related to one another, not just in three but also in higher dimensions (SN: 7/17/82, p. 42). Rourke and Rego's proof is based on an ingenious combination of handle theory and surgery.

One mathematician who has examined the proof closely is Wolfgan Haken of the University of Illinois at Urbana-Champaign. "It looks similar to things I tried 15 or so years ago," he says, "but there is one new idea." Using this idea, Haken tried to recreate the proof but failed. "I could not find a mistake," says Jaken, "but I could not confirm it [the proof] either."

Now Haken is waiting to see a more complete version of Rego and Rourke's proof. "Maybe there is a second new idea, which I did not realize is there," says Haken, "and it will work."

"It's hard to catch a subtle mistake," says Robion Kirby of the University of California at Berkeley, who has also started to study the proof. "I don't know how soon there'll be a definitive answer on this."
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Title Annotation:mathematics
Author:Peterson, Ivars
Publication:Science News
Date:Apr 5, 1986
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