By Вen Li
In science news this week, there is renewed interest in the Poincare Conjecture as news wires around the world picked up on a 16-month-old pre-print paper posted by a Russian scientist on-line.
According to a MathWorld Headline News release from April 15, 2003, “Poincare conjectured that the three-sphere is the only possible type of bounded three-dimensional space that contains no holes.” In simpler terms, the conjecture states that a given three-dimensional space is like a sphere (simply connected) if every closed loop that can be drawn on the object can be shrunk to a point.
The pre-print paper purporting to show this to be true, authored by Dr. Grigori Perelman, appears to have withstood more than a year of peer review. Dr. Perelman, part of the Steklov Institute of Mathematics (part of the Russian Academy of Sciences in St. Petersburg), lectured on his two-part, 61-page proof in the United States to great critical acclaim from his peers last year. If his proof remains unchallenged for two years after publication in a mathematics journal of world repute, he would be eligible for a $1 million prize offered by the Clay Mathematics Institute.
The Poincare Conjecture is one of seven Millennium Prize Problems, worth $1 million each, named by the Clay Mathematics Institute of Cambridge Massachusetts. According to several on-line discussions, Dr. Grigori has expressed little interest in collecting a $1 million prize for his work.
A number of solutions have been attempted since Poincare formulated his conjecture since 1904, most with fatal flaws discovered at the last minute. However, attempts to prove the deceptively simple conjecture have yielded valuable insights into low-dimensional topology.
This proof for the -sphere, if sustained, would be significant as it is the last remaining dimension of sphere for which the generalized form of the Poincare Conjecture has not been demonstrated. The 1 and 2 spheres are classical, the 4 sphere was proved in 1984, while a general proof for spheres of dimensions 5 and above was formulated in the 1960s, according to MathWorld.