Spheres in Disguise: Solid proof offered for famous conjecture

A Russian mathematician may have finally cracked one of the most famous problems in mathematics: the Poincaré conjecture, a question about the shapes of three-dimensional spaces. If his work is correct, it will make him eligible for a $1 million prize from the Clay Mathematics Institute in Cambridge, Mass., which has declared the conjecture one of the seven most important mathematical problems of the new millennium.

More than 100 mathematicians packed a lecture hall at the State University of New York at Stony Brook this week to hear Grigori Perelman of the Steklov Mathematical Institute in St. Petersburg, Russia, describe his work. Last week, Perelman told an equally attentive audience at the Massachusetts Institute of Technology (MIT) that he has proven the conjecture together with a broader problem called the Thurston geometrization conjecture. This second problem proposes that any three-dimensional space can be chopped in a standard way into pieces, each of which has a simple geometric structure.

Perelman has posted two papers about his research on the Internet (http://xxx.lanl.gov/abs/math.DG/0303109 and http://xxx.lanl.gov/abs/math.DG/0211159). Mathematicians are now scrutinizing every line of the work to verify its correctness.

“We’re all waiting with bated breath,” says Yair Minsky of Stony Brook, who attended the lecture there.

The Poincaré conjecture belongs to the field of topology, which studies properties that are preserved when a shape is stretched or twisted without tearing. Topologically speaking, the surfaces of a doughnut and of a coffee cup are the same, but they’re different from the surface of a ball.

Mathematicians have established criteria for distinguishing among types of surfaces. For example, consider a loop of string lying on a closed surface. More than a century ago, mathematicians proved that if every such loop can be shrunk to a single point without leaving the surface, the object is a sphere. On a doughnut, by contrast, a loop that encircles the hole can’t be shrunk to a point.

The traditional, or two-dimensional, sphere is the set of all points in three-dimensional space that are a given distance from a fixed center. Mathematicians also study what they call the three-dimensional sphere–the set of all points a given distance from a center in four-dimensional space. French mathematician Henri Poincaré conjectured 99 years ago that, just as in the case of surfaces, any closed three-dimensional space in which loops can be tightened to a single point is really a three-dimensional sphere.

In the intervening years, dozens of mathematicians have put forth mistaken proofs, often very subtly in error. For that reason, mathematicians are hesitant to declare the conjecture settled until Perelman’s proof has been thoroughly checked. But they agree that, unlike most previous attempts, Perelman’s papers contain a wealth of important ideas that will be valuable even if his work turns out to fall short of proving the full Poincaré conjecture.

“The first paper is already amazing,” says Jeff Viaclovsky of MIT. “It’s a major breakthrough.”


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