Grisha Perelman refuses to accept the Fields medal
Would Dr Grigory 'Grisha' Perelman, the reclusive math genius, who solved a central part of a century old puzzle called the Poincaire conjecture, attend the International Congress of Mathematicians, in Madrid, to receive his Fields medal?
The Fields medal is awarded in recognition of work in mathematics to recepients who are 40 years or younger. The Poincaire conjecture, stated by Henri Poincaire, in 1904, postulates that any shape that does not have any holes and fits within a finite space can be stretched and deformed into a sphere. That is true for any two dimensional surface in a everyday three dimensional world. The conjecture holds that this is true even for a three dimensional surface in a four or more dimensional world. The published proof to the Poincaire conjecture carries a one million $ prize awarded by the Clay Math Institute, Cambridge, Massachussettes.
As it turns out Grisha Perelman in keeping with his previous history of declining awards and offers of tenured positions from Princeton, Stanford, and other universities, was a no show at this years Fields award, adding to his already legendary status. Even the way he announced his proof - which took eight years to complete - was unusual. Rather than publishing in a peer-reviewed journal, he posted three manuscripts in an online archive of maths and physics papers. Dr.John Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman’s prescription to produce a more detailed 473-page step-by-step proof only of Poincaré’s Conjecture. “Perelman did all the work,” Dr. Morgan said. “This is just explaining it.”
In the late 1970’s, Dr. William Thurston of Cornell, extended Poincaré’s conjecture, showing that it was only a special case of a more powerful and general conjecture about three-dimensional geometry, namely that any space can be decomposed into a few basic shapes.
Dr Richard Hamilton at Columbia, set about investigating the Thurston conjecture, using the Ricci flow, a technique established by Hamilton, to investigate the underlying shape of an spaces borrowing mathematical concepts that underlie Einstein's theory of relativity and string theory. Hamilton's technique makes use of the fact that the metric, a quantitative measure, can be used to determine the distances between two nearby points in space. Using the mathematical technique of the Ricci flow to the metric, Hamilton was able to smooth away the bumps and curves of a surface, to reveal the underlying shape. Just like a hair dryer is used to shrink wrap plastic. It worked well on fairly round surfaces but in more complex surfaces with edges, pinches or kinks with infinite density, called singularities, would occur. The problem was then that the underlying surfaces of more complex surfaces became difficult to ascertain unless toplogists (scientists who study the shape of surfaces), 'surgically' removed these singularities.
It was Grisha Perelman again, who proved that these singularities were of no consequence and themselves turned into the shape of a sphere or tube in finite time after the Ricci flow began, and could be removed by toplogists to reveal the underlying spherical nature of most surfaces.
Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman’s proof of the two conjectures.
Little is known about Dr Perelman, who refuses to talk to the media. He was born on June 13, 1966 and his prodigious talent led to his early enrolment at a St Petersburg school specialising in advanced mathematics and physics. At the age of 16, he won a gold medal with a perfect score at the 1982 International Mathematical Olympiad, a competition for gifted schoolchildren. After receiving his PhD from the St Petersburg State University, he worked at the Steklov Institute of Mathematics before moving to the US in the late 80s to take posts at various universities. He returned to the Steklov about 10 years ago to work on his proof of the universe's shape.
Amongst his interests are walking in the woods outside the Steklov Institute of Mathematics in St.Petersburg, picking mushrooms. He is described as painfully shy and polite and has no interest in worldly or material wealth. No one knows when they will make contact with Dr Perelman as he steadfastly refuses to meet or keep email contact.
Articles by Grisha Perelman
1.The Entropy Formula for the Ricci Flow and its Geometric Applications, arXiv.org, November 11, 2002.
2. Ricci Flow with Surgery on Three-Manifolds, arXiv.org, March 10, 2003.
3. Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds, arXiv.org, July 17, 2003.
Detailed Expositions
1.A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow, by Huai-Dong Cao and Xi-Ping Zhu, Asian Journal of Mathematics, June 2006.
2. Notes on Perelman's Papers, by Bruce Kleiner and John Lott, arXiv.org, May 25, 2006.
3. Ricci Flow and the Poincaré Conjecture, by John Morgan and Gang Tian, arXiv.org, July 25, 2006.