[MLUG - DISCUSSION] Elusive Proof, Elusive Prover: A New Mathematical Mystery

[Mike Miller <EMAIL:PROTECTED>]

This is an amazing story.  The interesting part of it is that this Russian 
mathematician seems to have produced one of the most difficult proofs in 
the history of mathematics (now verified by top experts after several 
years of scrutiny) but he is uninterested in fame and fortune and they 
can't even find the guy.  He is a top genius, but he turns down prizes and 
job offers (e.g., professorship at Stanford).  He prefers to walk in the 
woods looking for mushrooms.  He doesn't pay attention to his appearance 
-- hair and beard are long and scruffy and he doesn't clip his 
fingernails.  Perhaps this is yet another example of how madness and 
genius often go hand in hand.  --Mike

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http://www.nytimes.com/2006/08/15/science/15math.html

N.Y. Times
August 15, 2006

Elusive Proof, Elusive Prover: A New Mathematical Mystery

By DENNIS OVERBYE

Correction Appended

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.

As a result there is a growing feeling, a cautious optimism that they have 
finally achieved a landmark not just of mathematics, but of human thought.

"It's really a great moment in mathematics," said Bruce Kleiner of Yale, 
who has spent the last three years helping to explicate Dr. Perelman's 
work. "It could have happened 100 years from now, or never."

In a speech at a conference in Beijing this summer, Shing-Tung Yau of 
Harvard said the understanding of three-dimensional space brought about by 
Poincaré's conjecture could be one of the major pillars of math in the 
21st century.

Quoting Poincaré himself, Dr.Yau said, "Thought is only a flash in the 
middle of a long night, but the flash that means everything."

But at the moment of his putative triumph, Dr. Perelman is nowhere in 
sight. He is an odds-on favorite to win a Fields Medal, math's version of 
the Nobel Prize, when the International Mathematics Union convenes in 
Madrid next Tuesday. But there is no indication whether he will show up.

Also left hanging, for now, is $1 million offered by the Clay Mathematics 
Institute in Cambridge, Mass., for the first published proof of the 
conjecture, one of seven outstanding questions for which they offered a 
ransom back at the beginning of the millennium.

"It's very unusual in math that somebody announces a result this big and 
leaves it hanging," said John Morgan of Columbia, one of the scholars who 
has also been filling in the details of Dr. Perelman's work.

Mathematicians have been waiting for this result for more than 100 years, 
ever since the French polymath Henri Poincaré posed the problem in 1904. 
And they acknowledge that it may be another 100 years before its full 
implications for math and physics are understood. For now, they say, it is 
just beautiful, like art or a challenging new opera.

Dr. Morgan said the excitement came not from the final proof of the 
conjecture, which everybody felt was true, but the method, "finding deep 
connections between what were unrelated fields of mathematics."

William Thurston of Cornell, the author of a deeper conjecture that 
includes Poincaré's and that is now apparently proved, said, "Math is 
really about the human mind, about how people can think effectively, and 
why curiosity is quite a good guide," explaining that curiosity is tied in 
some way with intuition.

"You don't see what you're seeing until you see it," Dr. Thurston said, 
"but when you do see it, it lets you see many other things."

Depending on who is talking, Poincaré's conjecture can sound either 
daunting or deceptively simple. It asserts that if any loop in a certain 
kind of three-dimensional space can be shrunk to a point without ripping 
or tearing either the loop or the space, the space is equivalent to a 
sphere.

The conjecture is fundamental to topology, the branch of math that deals 
with shapes, sometimes described as geometry without the details. To a 
topologist, a sphere, a cigar and a rabbit's head are all the same because 
they can be deformed into one another. Likewise, a coffee mug and a 
doughnut are also the same because each has one hole, but they are not 
equivalent to a sphere.

In effect, what Poincaré suggested was that anything without holes has to 
be a sphere. The one qualification was that this "anything" had to be what 
mathematicians call compact, or closed, meaning that it has a finite 
extent: no matter how far you strike out in one direction or another, you 
can get only so far away before you start coming back, the way you can 
never get more than 12,500 miles from home on the Earth.

In the case of two dimensions, like the surface of a sphere or a doughnut, 
it is easy to see what Poincaré was talking about: imagine a rubber band 
stretched around an apple or a doughnut; on the apple, the rubber band can 
be shrunk without limit, but on the doughnut it is stopped by the hole.

With three dimensions, it is harder to discern the overall shape of 
something; we cannot see where the holes might be. "We can't draw pictures 
of 3-D spaces," Dr. Morgan said, explaining that when we envision the 
surface of a sphere or an apple, we are really seeing a two-dimensional 
object embedded in three dimensions. Indeed, astronomers are still arguing 
about the overall shape of the universe, wondering if its topology 
resembles a sphere, a bagel or something even more complicated.

Poincaré's conjecture was subsequently generalized to any number of 
dimensions, but in fact the three-dimensional version has turned out to be 
the most difficult of all cases to prove. In 1960 Stephen Smale, now at 
the Toyota Technological Institute at Chicago, proved that it is true in 
five or more dimensions and was awarded a Fields Medal. In 1983, Michael 
Freedman, now at Microsoft, proved that it is true in four dimensions and 
also won a Fields.

"You get a Fields Medal for just getting close to this conjecture," Dr. 
Morgan said.

In the late 1970's, Dr. Thurston 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.

Mathematicians had known since the time of Georg Friedrich Bernhard 
Riemann, in the 19th century, that in two dimensions there are only three 
possible shapes: flat like a sheet of paper, closed like a sphere, or 
curved uniformly in two opposite directions like a saddle or the flare of 
a trumpet. Dr. Thurston suggested that eight different shapes could be 
used to make up any three-dimensional space.

"Thurston's conjecture almost leads to a list," Dr. Morgan said. "If it is 
true," he added, "Poincaré's conjecture falls out immediately." Dr. 
Thurston won a Fields in 1982.

Topologists have developed an elaborate set of tools to study and dissect 
shapes, including imaginary cutting and pasting, which they refer to as 
"surgery," but they were not getting anywhere for a long time.

In the early 1980's Richard Hamilton of Columbia suggested a new 
technique, called the Ricci flow, borrowed from the kind of mathematics 
that underlies Einstein's general theory of relativity and string theory, 
to investigate the shapes of spaces.

Dr. Hamilton's technique makes use of the fact that for any kind of 
geometric space there is a formula called the metric, which determines the 
distance between any pair of nearby points. Applied mathematically to this 
metric, the Ricci flow acts like heat, flowing through the space in 
question, smoothing and straightening all its bumps and curves to reveal 
its essential shape, the way a hair dryer shrink-wraps plastic.

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.

Dr. Perelman had already established himself as a master of differential 
geometry, the study of curves and surfaces, which is essential to, among 
other things, relativity and string theory Born in St. Petersburg in 1966, 
he distinguished himself as a high school student by winning a gold medal 
with a perfect score in the International Mathematical Olympiad in 1982. 
After getting a Ph.D.  from St. Petersburg State, he joined the Steklov 
Institute of Mathematics at St. Petersburg.

In a series of postdoctoral fellowships in the United States in the early 
1990's, Dr. Perelman impressed his colleagues as "a kind of unworldly 
person," in the words of Dr. Greene of U.C.L.A. -- friendly, but shy and 
not interested in material wealth.

"He looked like Rasputin, with long hair and fingernails," Dr. Greene 
said.

Asked about Dr. Perelman's pleasures, Dr. Anderson said that he talked a 
lot about hiking in the woods near St. Petersburg looking for mushrooms.

Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, 
spurning offers from Stanford and Princeton, among others. In 1996 he 
added to his legend by turning down a prize for young mathematicians from 
the European Mathematics Society.

Until his papers on Poincaré started appearing, some friends thought Dr. 
Perelman had left mathematics. Although they were so technical and 
abbreviated that few mathematicians could read them, they quickly 
attracted interest among experts. In the spring of 2003, Dr. Perelman came 
back to the United States to give a series of lectures at Stony Brook and 
the Massachusetts Institute of Technology, and also spoke at Columbia, New 
York University and Princeton.

But once he was back in St. Petersburg, he did not respond to further 
invitations. The e-mail gradually ceased.

"He came once, he explained things, and that was it," Dr. Anderson said. 
"Anything else was superfluous."

Recently, Dr. Perelman is said to have resigned from Steklov. E-mail 
messages addressed to him and to the Steklov Institute went unanswered.

In his absence, others have taken the lead in trying to verify and 
disseminate his work. 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.

Dr. 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."

Both works were supported by the Clay institute, which has posted them on 
its Web site, claymath.org . Meanwhile, Huai-Dong Cao of Lehigh University 
and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have 
published their own 318-page proof of both conjectures in The Asian 
Journal of Mathematics (www.ims.cuhk.edu.hk/).

Although these works were all hammered out in the midst of discussion and 
argument by experts, in workshops and lectures, they are about to receive 
even stricter scrutiny and perhaps crossfire. "Caution is appropriate," 
said Dr. Kleiner, because the Poincaré conjecture is not just famous, but 
important.

James Carlson, president of the Clay Institute, said the appearance of 
these papers had started the clock ticking on a two-year waiting period 
mandated by the rules of the Clay Millennium Prize. After two years, he 
said, a committee will be appointed to recommend a winner or winners if it 
decides the proof has stood the test of time.

"There is nothing in the rules to prevent Perelman from receiving all or 
part of the prize," Dr. Carlson said, saying that Dr. Perelman and Dr. 
Hamilton had obviously made the main contributions to the proof.

In a lecture at M.I.T. in 2003, Dr. Perelman described himself "in a way" 
as Dr. Hamilton's disciple, although they had never worked together. Dr. 
Hamilton, who got his Ph.D. from Princeton in 1966, is too old to win the 
Fields medal, which is given only up to the age of 40, but he is slated to 
give the major address about the Poincaré conjecture in Madrid next week. 
He did not respond to requests for an interview.

Allowing that Dr. Perelman, should he win the Clay Prize, might refuse the 
honor, Dr. Carlson said the institute could decide instead to use award 
money to support Russian mathematicians, the Steklov Institute or even the 
Math Olympiad.

Dr. Anderson said that to some extent the new round of papers already 
represented a kind of peer review of Dr. Perelman's work. "All these 
together make the case pretty clear," he said. "The community accepts the 
validity of his work. It's commendable that the community has gotten 
together."


Copyright 2006 The New York Times Company

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