Elusive Proof, Elusive Prover: A New Mathematical Mystery
By DENNIS OVERBYE
Published: August 15, 2006
Grisha Perelman, where are you?
Three years ago, a Russian mathematician by the name of Grigory
Perelman, a k a Grisha, in St. Petersburg, announced that he had solved
a famous and intractable mathematical problem, known as the Poincaré
conjecture, about the nature of space.
After posting a few short papers on the Internet and making a whirlwind
lecture tour of the United States, Dr. Perelman disappeared back into
the Russian woods in the spring of 2003, leaving the world’s
mathematicians to pick up the pieces and decide if he was right.
Now they say they have finished his work, and the evidence is
circulating among scholars in the form of three book-length papers with
about 1,000 pages of dense mathematics and prose between them.
....
Dr. Hamilton succeeded in showing that certain generally round objects,
like a head, would evolve into spheres under this process, but the fates
of more complicated objects were problematic. As the Ricci flow
progressed, kinks and neck pinches, places of infinite density known as
singularities, could appear, pinch off and even shrink away. Topologists
could cut them away, but there was no guarantee that new ones would not
keep popping up forever.
“All sorts of things can potentially happen in the Ricci flow,” said
Robert Greene, a mathematician at the University of California, Los
Angeles. Nobody knew what to do with these things, so the result was a
logjam.
It was Dr. Perelman who broke the logjam. He was able to show that the
singularities were all friendly. They turned into spheres or tubes.
Moreover, they did it in a finite time once the Ricci flow started. That
meant topologists could, in their fashion, cut them off, and allow the
Ricci process to continue to its end, revealing the topologically
spherical essence of the space in question, and thus proving the
conjectures of both Poincaré and Thurston.
Dr. Perelman’s first paper, promising “a sketch of an eclectic proof,”
came as a bolt from the blue when it was posted on the Internet in
November 2002. “Nobody knew he was working on the Poincaré conjecture,”
said Michael T. Anderson of the State University of New York in Stony Brook.
cont'd....
http://www.nytimes.com/2006/08/15/science/15math.html?8dpc=&_r=1&pagewanted=all
and....
http://www.claymath.org/millennium/Poincare_Conjecture/