Mathematical Digest

Short Summaries of Articles about Mathematics
in the Popular Press

``$1 million mathematical mystery `solved','' by Robert Matthews., 9 April 2002.
``Die Millionen-Frage,'' wire service report. Suüddeutsche Zeitung, 16 April 2002, page V2/11.
"British brain aims to grab maths millions," Nature, 18 April 2002.
"UK Math Wiz May Have Solved Problem," Associated Press. New York Times, 25 April 2002.

These three stories report on a purported solution of the famous and longstanding problem known as the Poincaré Conjecture. Named after the French mathematician Henri Poincaré (1854-1912), the conjecture asks a fundamental question about three-dimensional shapes. To understand what the conjecture is about, it's easiest to start with the analogous question for two-dimensional surfaces, such as the sphere: If any loop lying on a two-dimensional surface can be shrunk (without cutting) to a point, is that surface fundamentally the same as a sphere? The answer for two-dimensional surfaces is yes. The surface of a doughnut (called a torus in mathematics) is a two-dimensional surface such that not all loops can be shrunk to points; thus a torus is fundamentally different from a sphere. The analogous question can be posed for shapes of any dimension, and indeed the question has been answered all dimensions except dimension three. The question, when posed in three dimensions, is called the Poincaré Conjecture. Mathematicians have for decades tried to prove the conjecture, without success. Now Martin Dunwoody of Southampton University in England has posted on his web site what he hopes is a correct proof of the Poincaré Conjecture. At the time of this writing, other mathematicians were checking Dunwoody's proof, and many were skeptical it would hold up. But if it does, Dunwoody would be in line for a US$1 million prize offered by the Clay Mathematics Institute.

--- Allyn Jackson