Elusive Proof, Elusive Prover: A New Mathematical
Mystery
By Dennis Overbye
NYT 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 booklength 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, ShingTung Yau of
Harvard said the understanding of threedimensional 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 oddson 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 threedimensional 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 3D spaces,” Dr. Morgan said, explaining that
when we envision the surface of a sphere or an apple, we are really
seeing a twodimensional 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 threedimensional 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 threedimensional 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 threedimensional 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 shrinkwraps 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 email 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. Email
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 473page stepbystep
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, HuaiDong Cao of Lehigh University
and XiPing Zhu of Zhongshan University in Guangzhou, China, have published
their own 318page proof of both conjectures in The Asian Journal of
Mathematics.
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 twoyear 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.”
