Blog on Math Blogs

Mathematical Digest

Short Summaries of Articles about Mathematics
in the Popular Press

"Celebrated Math Problem Solved, Russian Reports," by Sara Robinson. New York Times, 15 April 2003.
"Mathematics World Abuzz Over Possible Poincaré," by Dana Mackenzie. Science, 18 April 2003.
"A Mathematician's World of Doughnuts and Spheres," by George Johnson. New York Times, 20 April 2003.
"Poincaré proof adds up to potential payday." Nature, 24 April 2003.
"Spheres in Disguise," by Erica Klarreich. Science News, 26 April 2003.
"Poincaré Solved?," by Jenny Hogan. New Scientist, 26 April 2003, page 8;
"If it looks like a sphere...," by Erica Klarreich. Science News, 14 June 2003 (click here for a Math Digest of this article).

These articles discuss reports that a Russian mathematician, Grigory Perelman, has solved the Poincaré Conjecture. Long one of the outstanding problems in mathematics, the Poincaré Conjecture is one of the seven "Millennium Prize Problems" for which the Clay Mathematics Institute has offered a prize of US$1 million.

The conjecture was first proposed in the early 20th century by the French mathematician Henri Poincaré. It concerns fundamental properties of shapes such as a sphere, a ball, or the surface of a doughnut---called in math lingo a "torus". Spheres and toruses are two-dimensional shapes; the interior of a ball is a three-dimensional shape. Mathematicians also routinely deal with higher-dimensional counterparts of these shapes. In its two-dimensional version, the Poincaré Conjecture says, essentially, that any two-dimensional surface without holes is a sphere---it might have have a few bends or twists in it, but fundamentally it is a sphere. This version of the conjecture has been known to be true since the 19th century. During the 20th century, versions of the conjecture for dimensions 4 and higher were proved. The one case that remains unproved is the one Poincaré originally proposed, namely, the case of 3 dimensions.

The news reports about Perelman's work have expressed cautious optimism. Experts in the field are still examining his work to see if all the details are right and whether the proof holds together. The New York Times article by Sara Robinson contains quotations of MIT mathematician Tomasz Mrowka that sum up the general feeling among mathematicians: "It's not certain, but we're taking it very seriously. [Perelman has] obviously thought about this stuff very hard for a long time, and it will be very hard to find any mistakes." The article also quotes Mrowka as saying: "This is one of those happy circumstances where it's going to be fun no matter what. Either he's done it or he's made some really significant progress, and we're going to learn from it."

--- Allyn Jackson

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