Grisha Perelman, where are you?

Elusive proof, elusive prover: A new math mystery

By Dennis Overbye The New York Times
Published: August 15, 2006

Grisha Perelman, where are you?

Three years ago, a Russian mathematician by the name of Grigory Perelman 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, Perelman, known as Grisha, disappeared
back into the Russian woods in the spring of 2003, leaving the world's
mathematicians to pick up the pieces and decide whether 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
University, who has spent the last three years helping to explicate
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
the Poincaré conjecture could be one of the major pillars of math in the
21st century.

But at the moment of his putative triumph, 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 the $1 million offered by the Clay
Mathematics Institute in Cambridge, Massachusetts, for the first published
proof of the conjecture, one of seven outstanding questions for which they
offered a ransom 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 University in New York, one
of the scholars who has also been filling in the details of 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.

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 University, the author of a deeper conjecture,
which includes Poincaré's and which 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," saying that curiosity is tied in some way
with intuition.

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

Depending on who is talking, the Poincaré 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, or 20,100 kilometers, from home on 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.

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
Anderson of the State University of New York in Stony Brook.

Perelman had already established himself as a master of differential
geometry, the study of curves and surfaces. 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 doctorate 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
1990s, Perelman impressed his colleagues as "a kind of unworldly person," in
the words of Robert Greene, a mathematician at the University of California,
Los Angeles - friendly, but shy and not interested in material wealth.

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

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

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 the Poincaré conjecture started appearing, some friends
thought 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, Perelman came back to the
United States to give a series of lectures.

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," Anderson said.
"Anything else was superfluous."

Recently, 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.


Grisha Perelman, where are you?

Three years ago, a Russian mathematician by the name of Grigory Perelman 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, Perelman, known as Grisha, disappeared
back into the Russian woods in the spring of 2003, leaving the world's
mathematicians to pick up the pieces and decide whether 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
University, who has spent the last three years helping to explicate
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
the Poincaré conjecture could be one of the major pillars of math in the
21st century.

But at the moment of his putative triumph, 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 the $1 million offered by the Clay
Mathematics Institute in Cambridge, Massachusetts, for the first published
proof of the conjecture, one of seven outstanding questions for which they
offered a ransom 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 University in New York, one
of the scholars who has also been filling in the details of 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.

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 University, the author of a deeper conjecture,
which includes Poincaré's and which 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," saying that curiosity is tied in some way
with intuition.

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

Depending on who is talking, the Poincaré 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, or 20,100 kilometers, from home on 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.

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
Anderson of the State University of New York in Stony Brook.

Perelman had already established himself as a master of differential
geometry, the study of curves and surfaces. 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 doctorate 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
1990s, Perelman impressed his colleagues as "a kind of unworldly person," in
the words of Robert Greene, a mathematician at the University of California,
Los Angeles - friendly, but shy and not interested in material wealth.

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

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

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 the Poincaré conjecture started appearing, some friends
thought 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, Perelman came back to the
United States to give a series of lectures.

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," Anderson said.
"Anything else was superfluous."

Recently, 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.


.

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