Mathematical Digest

Short Summaries of Articles about Mathematics
in the Popular Press

"The Shapes of Space," by Graham P. Collins. Scientific American, July 2004, pages 94-103.

This article describes the achievement of Grigory Perelman, who appears to have established two outstanding results in mathematics, the Poincaré Conjecture and the Thurston Geometrization Conjecture. To understand what it's all about requires a bit of mathematical background, which Collins provides in the course of the article. The Poincaré Conjecture, first proposed in 1904 by the mathematician and physicist Henri Poincaré, asks, roughly speaking, whether the three-dimensional sphere is the simplest of all three-dimensional shapes. As a caption to the article notes: "Beware: the three-dimensional sphere is probably not what you think it is!" A one-dimensional sphere is a circle, which is the set of all points equidistant from a given point in the plane; a two-dimensional sphere is the surface of a ball, and this surface is the set of all points equidistant from a given point in three dimensions. A three-dimensional sphere is defined in the same way, as the set of points equidistant from a point in four dimensions. The article helps the reader through some abstract mental gymnastics to get acclimated to these ideas. Perelman showed that, just as Poincaré speculated a century ago, the three-dimensional sphere is the simplest kind of three-dimensional shape there is. The Thurston Geometrization Conjecture was first laid out by William P. Thurston in the 1970s. This conjecture encapsulates a far-reaching vision of how to use geometry to characterize three-dimensional shapes. The Thurston Geometrization Conjecture actually subsumes the Poincaré Conjecture as a special case. Perelman posted the papers containing his proofs of the two conjectures on the arXiv, a preprint server widely used by mathematicians; he did not submit the papers to a journal, which would have been the traditional route. Nevertheless, his papers have been widely read by mathematicians all over the world, many of whom have pored over them with great care. No major mistakes have been found in his proofs, but at the same time there is not an obvious consensus among mathematicians that the proofs are correct. More time is needed for such a consensus to clearly emerge.

Other articles on Perelman's work:

"Taming the Fourth Dimension," by Bruce Schecter. New Scientist, 17 July 2004, pages 26-29.
"Million-dollar problem solved." New Scientist, 20/27 December 2003, page 22.
"Century-old math problem may have been solved," by Jascha Hoffman. Boston Globe, 30 December 2003.
"2003 in context: Highlights," Nature, 18/25 December 2003.

See also the Math Digest summary of earlier articles on this same topic:

Articles on the attempt to prove the Poincaré Conjecture, April 2003

--- Allyn Jackson