A New Mathematical Mystery

Elusive Proof, Elusive Prover: A New Mathematical Mystery
Xianfeng David Gu and Shing-Tung Yau
Even topologists don?t think this soap film can be made into a sphere.

By DENNIS OVERBYE
Published: August 15, 2006
Grisha Perelman, where are you?


Xianfeng David Gu and Shing-Tung Yau
To a topologist, a rabbit is the same as a sphere. Neither has a hole. Longitude
and latitude lines on the rabbit allow mathematicians to map it onto different
forms while preserving information.
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.

Page 2 of 3)



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.

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Top, Hulton-Deutsch Collection/Corbis; center, Bill Wingell for The New York
Times; above, Allison Evans/Clay Mathematics Institute
THE MATHEMATICIANS Henri Poincaré, top, posed his vexing problem in 1904. In
1986, William Thurston, center, of Cornell won a Fields Medal for expanding on
it. Richard Hamilton, above, of Columbia invented a way to help solve it.

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Graphic: The Essential Grisha 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 *** 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 1986.

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.

Elusive Proof, Elusive Prover: A New Mathematical Mystery
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Published: August 15, 2006
(Page 3 of 3)



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.

Skip to next paragraph
Ask Science
Dennis Overbye will answer select reader questions about this article. Answers
will be posted on nytimes.com/science later in the week.

More Ask Science »
Related Web Link
Clay Mathematics Institute
Multimedia

Graphic: The Essential Grisha ?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.?

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