Tuesday, August 22, 2006

A Russian mathematician, Grigori (“Grisha”) Perelman, who is credited proving the Poincaré conjecture declined to accept the Fields Medal, regarded as the highest honor in the field of mathematics.

The Fields Medal, often referred to as the “Nobel Prize of mathematics”, was awarded this year to Andrei Okounkov (Russia/US), Terence Tao (Australia/US) and Wendelin Werner (France) in addition to Perelman. The award was handed out by King Juan Carlos of Spain and is accompanied by a C$15’000 (approximately US$13’400 or 10’400EUR) cash prize (less than the one million Euros that come with the Nobel prize). Nominees have to be under 40 years, because the founder of the award, Canadian mathematician John Charles Fields wanted the medal to be a stimulus for future endeavours.

Perelman submitted two papers in 2002 and 2003 outlining a proof for Thurston’s geometrization conjecture, which in turn, implies a proof for the Poincaré conjecture. Other mathematicians filling in the details have found no flaws in Perelman’s approach yet. In 2003, Perelman made a short tour in the United States to explain his proof of the conjecture. When he went back to the St Petersburg department of the Steklov Institute of Mathematics, he gave up his job, and is reported to be unemployed and living with his mother ever since.

“The reason Perelman gave me is that he feels isolated from the mathematical community and therefore has no wish to appear as one of its leaders.” declared Manuel de Leon, chairman of the Congress, when asked about Grisha’s motivation to decline. Prof. John Ball, retiring president of the International Mathematical Union, added: “The reason centres on his feeling of isolation from the mathematical community.” Perelman’s friend Anatoly Vershik said the reclusive math genius just wanted to be declared correct, and regarded recognition as superficial.

The Poincaré conjecture is widely considered one of the most important questions in topology (a branch of mathematics concerned with spatial properties preserved under deformation like stretching without tearing or gluing). It is one of the seven Millennium Prize Problems for which the Clay Mathematics Institute is offering a $1,000,000 prize for a correct solution.

Observers speculate that he will also refuse this prize. In 1997, he also refused an award by the European Congress of Mathematicians, because he deemed the judges unable to understand his work. A spokesman for the Institute said it would decide on the prize in two years. Richard Hamilton’sRicci flow” equation could also earn him a part of the prize, since it formed the basis for Perelman’s papers.

In 1966, German Alexander Grothendieck refused his Fields Medal in Moscow out of protest against the presence of the Red Army in Eastern Europe. But later he accepted it.

The Poincaré conjecture is not an easy thing to explain in plain English. A sphere as we know it is called a two-dimensional sphere in topology (because it’s surface can be approximated by a two-dimensional plane). The four-dimensional analogue of such a 2-sphere is a 3-sphere (which is an example of a 3-manifold).

Now imagine a ball and a donut made of rubber. If you throw a lasso around the ball and pull, you can squeeze it to a single point and slide off the lasso. But you can’t do that with the noose through the hole of a donut, the only way is to cut through the donut. So in topology, there are basically two kinds of objects: objects with or without holes. By deforming objects without holes you can make them look like a sphere, but this is impossible for objects with holes. So basically, Grisha proved that in topology a ball and a banana are the same.

The Poincaré conjecture surmises that if a closed three-dimensional manifold (our multiple-dimension banana) is sufficiently like a 3-sphere (a kind of hypersphere) in that each loop in the manifold can be tightened to a point, then it is really just a three-dimensional sphere.