## Manifold Destiny
## A legendary problem and the battle over who solved itBy Sylvia Nasar and David Gruber August 28, 2006 Issue - The New Yorker |
Grigory Perelman (right) says: "If the proof is correct, thenno other recognition is needed." Shing-Tung Yau isn't so sure... PIERRE LE-TAN |

On the evening of June 20th, several hundred physicists, including a
Nobel laureate, assembled in an auditorium at the Friendship Hotel in
Beijing for a lecture by the Chinese mathematician Shing-Tung Yau. In
the late nineteen-seventies, when Yau was in his twenties, he had made
a series of breakthroughs that helped launch the string-theory
revolution in physics and earned him, in addition to a Fields
Medal--the most coveted award in mathematics--a reputation in both
disciplines as a thinker of unrivalled technical power.

Yau had since become a professor of mathematics at Harvard and the
director of mathematics institutes in Beijing and Hong Kong, dividing
his time between the United States and China. His lecture at the
Friendship Hotel was part of an international conference on string
theory, which he had organized with the support of the Chinese
government, in part to promote the country's recent advances in
theoretical physics. (More than six thousand students attended the
keynote address, which was delivered by Yau's close friend Stephen
Hawking, in the Great Hall of the People.) The subject of Yau's talk
was something that few in his audience knew much about: the Poincaré
conjecture, a century-old conundrum about the characteristics of
three-dimensional spheres, which, because it has important implications
for mathematics and cosmology and because it has eluded all attempts at
solution, is regarded by mathematicians as a holy grail.

Yau, a stocky man of fifty-seven, stood at a lectern in shirtsleeves
and black-rimmed glasses and, with his hands in his pockets, described
how two of his students, Xi-Ping Zhu and Huai-Dong Cao, had completed a
proof of the Poincaré conjecture a few weeks earlier. "I'm very
positive about Zhu and Cao's work," Yau said. "Chinese mathematicians
should have every reason to be proud of such a big success in
completely solving the puzzle." He said that Zhu and Cao were indebted
to his longtime American collaborator Richard Hamilton, who deserved
most of the credit for solving the Poincaré. He also mentioned Grigory
Perelman, a Russian mathematician who, he acknowledged, had made an
important contribution. Nevertheless, Yau said, "in Perelman's work,
spectacular as it is, many key ideas of the proofs are sketched or
outlined, and complete details are often missing." He added, "We would
like to get Perelman to make comments. But Perelman resides in St.
Petersburg and refuses to communicate with other people."

For ninety minutes, Yau discussed some of the technical details of his
students' proof. When he was finished, no one asked any questions. That
night, however, a Brazilian physicist posted a report of the lecture on
his blog. "Looks like China soon will take the lead also in
mathematics," he wrote.

Grigory Perelman is indeed reclusive. He left his job as a researcher
at the Steklov Institute of Mathematics, in St. Petersburg, last
December; he has few friends; and he lives with his mother in an
apartment on the outskirts of the city. Although he had never granted
an interview before, he was cordial and frank when we visited him, in
late June, shortly after Yau's conference in Beijing, taking us on a
long walking tour of the city. "I'm looking for some friends, and they
don't have to be mathematicians," he said. The week before the
conference, Perelman had spent hours discussing the Poincaré conjecture
with Sir John M. Ball, the fifty-eight-year-old president of the
International Mathematical Union, the discipline's influential
professional association. The meeting, which took place at a conference
center in a stately mansion overlooking the Neva River, was highly
unusual. At the end of May, a committee of nine prominent
mathematicians had voted to award Perelman a Fields Medal for his work
on the Poincaré, and Ball had gone to St. Petersburg to persuade him to
accept the prize in a public ceremony at the I.M.U.'s quadrennial
congress, in Madrid, on August 22nd.

The Fields Medal, like the Nobel Prize, grew, in part, out of a desire
to elevate science above national animosities. German mathematicians
were excluded from the first I.M.U. congress, in 1924, and, though the
ban was lifted before the next one, the trauma it caused led, in 1936,
to the establishment of the Fields, a prize intended to be "as purely
international and impersonal as possible."

However, the Fields Medal, which is awarded every four years, to
between two and four mathematicians, is supposed not only to reward
past achievements but also to stimulate future research; for this
reason, it is given only to mathematicians aged forty and younger. In
recent decades, as the number of professional mathematicians has grown,
the Fields Medal has become increasingly prestigious. Only forty-four
medals have been awarded in nearly seventy years--including three for
work closely related to the Poincaré conjecture--and no mathematician
has ever refused the prize. Nevertheless, Perelman told Ball that he
had no intention of accepting it. "I refuse," he said simply.

Over a period of eight months, beginning in November, 2002, Perelman
posted a proof of the Poincaré on the Internet in three installments.
Like a sonnet or an aria, a mathematical proof has a distinct form and
set of conventions. It begins with axioms, or accepted truths, and
employs a series of logical statements to arrive at a conclusion. If
the logic is deemed to be watertight, then the result is a theorem.
Unlike proof in law or science, which is based on evidence and
therefore subject to qualification and revision, a proof of a theorem
is definitive. Judgments about the accuracy of a proof are mediated by
peer-reviewed journals; to insure fairness, reviewers are supposed to
be carefully chosen by journal editors, and the identity of a scholar
whose pa-per is under consideration is kept secret. Publication implies
that a proof is complete, correct, and original.

By these standards, Perelman's proof was unorthodox. It was
astonishingly brief for such an ambitious piece of work; logic
sequences that could have been elaborated over many pages were often
severely compressed. Moreover, the proof made no direct mention of the
Poincaré and included many elegant results that were irrelevant to the
central argument. But, four years later, at least two teams of experts
had vetted the proof and had found no significant gaps or errors in it.
A consensus was emerging in the math community: Perelman had solved the
Poincaré. Even so, the proof's complexity--and Perelman's use of
shorthand in making some of his most important claims--made it
vulnerable to challenge. Few mathematicians had the expertise necessary
to evaluate and defend it.

After giving a series of lectures on the proof in the United States in
2003, Perelman returned to St. Petersburg. Since then, although he had
continued to answer queries about it by e-mail, he had had minimal
contact with colleagues and, for reasons no one understood, had not
tried to publish it. Still, there was little doubt that Perelman, who
turned forty on June 13th, deserved a Fields Medal. As Ball planned the
I.M.U.'s 2006 congress, he began to conceive of it as a historic event.
More than three thousand mathematicians would be attending, and King
Juan Carlos of Spain had agreed to preside over the awards ceremony.
The I.M.U.'s newsletter predicted that the congress would be remembered
as "the occasion when this conjecture became a theorem." Ball,
determined to make sure that Perelman would be there, decided to go to
St. Petersburg.

Ball wanted to keep his visit a secret--the names of Fields Medal
recipients are announced officially at the awards ceremony--and the
conference center where he met with Perelman was deserted. For ten
hours over two days, he tried to persuade Perelman to agree to accept
the prize. Perelman, a slender, balding man with a curly beard, bushy
eyebrows, and blue-green eyes, listened politely. He had not spoken
English for three years, but he fluently parried Ball's entreaties, at
one point taking Ball on a long walk--one of Perelman's favorite
activities. As he summed up the conversation two weeks later: "He
proposed to me three alternatives: accept and come; accept and don't
come, and we will send you the medal later; third, I don't accept the
prize. From the very beginning, I told him I have chosen the third
one." The Fields Medal held no interest for him, Perelman explained.
"It was completely irrelevant for me," he said. "Everybody understood
that if the proof is correct then no other recognition is needed."

Proofs of the Poincaré have been announced nearly every year since the
conjecture was formulated, by Henri Poincaré, more than a hundred years
ago. Poincaré was a cousin of Raymond Poincaré, the President of France
during the First World War, and one of the most creative mathematicians
of the nineteenth century. Slight, myopic, and notoriously
absent-minded, he conceived his famous problem in 1904, eight years
before he died, and tucked it as an offhand question into the end of a
sixty-five-page paper.

Poincaré didn't make much progress on proving the conjecture. "Cette
question nous entraînerait trop loin" ("This question would take us too
far"), he wrote. He was a founder of topology, also known as
"rubber-sheet geometry," for its focus on the intrinsic properties of
spaces. From a topologist's perspective, there is no difference between
a bagel and a coffee cup with a handle. Each has a single hole and can
be manipulated to resemble the other without being torn or cut.
Poincaré used the term "manifold" to describe such an abstract
topological space. The simplest possible two-dimensional manifold is
the surface of a soccer ball, which, to a topologist, is a sphere--even
when it is stomped on, stretched, or crumpled. The proof that an object
is a so-called two-sphere, since it can take on any number of shapes,
is that it is "simply connected," meaning that no holes puncture it.
Unlike a soccer ball, a bagel is not a true sphere. If you tie a
slipknot around a soccer ball, you can easily pull the slipknot closed
by sliding it along the surface of the ball. But if you tie a slipknot
around a bagel through the hole in its middle you cannot pull the
slipknot closed without tearing the bagel.

Two-dimensional manifolds were well understood by the mid-nineteenth
century. But it remained unclear whether what was true for two
dimensions was also true for three. Poincaré proposed that all closed,
simply connected, three-dimensional manifolds--those which lack holes
and are of finite extent--were spheres. The conjecture was potentially
important for scientists studying the largest known three-dimensional
manifold: the universe. Proving it mathematically, however, was far
from easy. Most attempts were merely embarrassing, but some led to
important mathematical discoveries, including proofs of Dehn's Lemma,
the Sphere Theorem, and the Loop Theorem, which are now fundamental
concepts in topology.

By the nineteen-sixties, topology had become one of the most productive
areas of mathematics, and young topologists were launching regular
attacks on the Poincaré. To the astonishment of most mathematicians, it
turned out that manifolds of the fourth, fifth, and higher dimensions
were more tractable than those of the third dimension. By 1982,
Poincaré's conjecture had been proved in all dimensions except the
third. In 2000, the Clay Mathematics Institute, a private foundation
that promotes mathematical research, named the Poincaré one of the
seven most important outstanding problems in mathematics and offered a
million dollars to anyone who could prove it.

"My whole life as a mathematician has been dominated by the Poincaré
conjecture," John Morgan, the head of the mathematics department at
Columbia University, said. "I never thought I'd see a solution. I
thought nobody could touch it."

Grigory Perelman did not plan to become a mathematician. "There was
never a decision point," he said when we met. We were outside the
apartment building where he lives, in Kupchino, a neighborhood of drab
high-rises. Perelman's father, who was an electrical engineer,
encouraged his interest in math. "He gave me logical and other math
problems to think about," Perelman said. "He got a lot of books for me
to read. He taught me how to play chess. He was proud of me." Among the
books his father gave him was a copy of "Physics for Entertainment,"
which had been a best-seller in the Soviet Union in the
nineteen-thirties. In the foreword, the book's author describes the
contents as "conundrums, brain-teasers, entertaining anecdotes, and
unexpected comparisons," adding, "I have quoted extensively from Jules
Verne, H. G. Wells, Mark Twain and other writers, because, besides
providing entertainment, the fantastic experiments these writers
describe may well serve as instructive illustrations at physics
classes." The book's topics included how to jump from a moving car, and
why, "according to the law of buoyancy, we would never drown in the
Dead Sea."

The notion that Russian society considered worthwhile what Perelman did
for pleasure came as a surprise. By the time he was fourteen, he was
the star performer of a local math club. In 1982, the year that
Shing-Tung Yau won a Fields Medal, Perelman earned a perfect score and
the gold medal at the International Mathematical Olympiad, in Budapest.
He was friendly with his teammates but not close--"I had no close
friends," he said. He was one of two or three Jews in his grade, and he
had a passion for opera, which also set him apart from his peers. His
mother, a math teacher at a technical college, played the violin and
began taking him to the opera when he was six. By the time Perelman was
fifteen, he was spending his pocket money on records. He was thrilled
to own a recording of a famous 1946 performance of "La Traviata,"
featuring Licia Albanese as Violetta. "Her voice was very good," he
said.

At Leningrad University, which Perelman entered in 1982, at the age of
sixteen, he took advanced classes in geometry and solved a problem
posed by Yuri Burago, a mathematician at the Steklov Institute, who
later became his Ph.D. adviser. "There are a lot of students of high
ability who speak before thinking," Burago said. "Grisha was different.
He thought deeply. His answers were always correct. He always checked
very, very carefully." Burago added, "He was not fast. Speed means
nothing. Math doesn't depend on speed. It is about deep."

At the Steklov in the early nineties, Perelman became an expert on the
geometry of Riemannian and Alexandrov spaces--extensions of traditional
Euclidean geometry--and began to publish articles in the leading
Russian and American mathematics journals. In 1992, Perelman was
invited to spend a semester each at New York University and Stony Brook
University. By the time he left for the United States, that fall, the
Russian economy had collapsed. Dan Stroock, a mathematician at M.I.T.,
recalls smuggling wads of dollars into the country to deliver to a
retired mathematician at the Steklov, who, like many of his colleagues,
had become destitute.

Perelman was pleased to be in the United States, the capital of the
international mathematics community. He wore the same brown corduroy
jacket every day and told friends at N.Y.U. that he lived on a diet of
bread, cheese, and milk. He liked to walk to Brooklyn, where he had
relatives and could buy traditional Russian brown bread. Some of his
colleagues were taken aback by his fingernails, which were several
inches long. "If they grow, why wouldn't I let them grow?" he would say
when someone asked why he didn't cut them. Once a week, he and a young
Chinese mathematician named Gang Tian drove to Princeton, to attend a
seminar at the Institute for Advanced Study.

For several decades, the institute and nearby Princeton University had
been centers of topological research. In the late seventies, William
Thurston, a Princeton mathematician who liked to test out his ideas
using scissors and construction paper, proposed a taxonomy for
classifying manifolds of three dimensions. He argued that, while the
manifolds could be made to take on many different shapes, they
nonetheless had a "preferred" geometry, just as a piece of silk draped
over a dressmaker's mannequin takes on the mannequin's form.

Thurston proposed that every three-dimensional manifold could be broken
down into one or more of eight types of component, including a
spherical type. Thurston's theory--which became known as the
geometrization conjecture--describes all possible three-dimensional
manifolds and is thus a powerful generalization of the Poincaré. If it
was confirmed, then Poincaré's conjecture would be, too. Proving
Thurston and Poincaré "definitely swings open doors," Barry Mazur, a
mathematician at Harvard, said. The implications of the conjectures for
other disciplines may not be apparent for years, but for mathematicians
the problems are fundamental. "This is a kind of twentieth-century
Pythagorean theorem," Mazur added. "It changes the landscape."

In 1982, Thurston won a Fields Medal for his contributions to topology.
That year, Richard Hamilton, a mathematician at Cornell, published a
paper on an equation called the Ricci flow, which he suspected could be
relevant for solving Thurston's conjecture and thus the Poincaré. Like
a heat equation, which describes how heat distributes itself evenly
through a substance--flowing from hotter to cooler parts of a metal
sheet, for example--to create a more uniform temperature, the Ricci
flow, by smoothing out irregularities, gives manifolds a more uniform
geometry.

Hamilton, the son of a Cincinnati doctor, defied the math profession's
nerdy stereotype. Brash and irreverent, he rode horses, windsurfed, and
had a succession of girlfriends. He treated math as merely one of
life's pleasures. At forty-nine, he was considered a brilliant
lecturer, but he had published relatively little beyond a series of
seminal articles on the Ricci flow, and he had few graduate students.
Perelman had read Hamilton's papers and went to hear him give a talk at
the Institute for Advanced Study. Afterward, Perelman shyly spoke to
him.

"I really wanted to ask him something," Perelman recalled. "He was
smiling, and he was quite patient. He actually told me a couple of
things that he published a few years later. He did not hesitate to tell
me. Hamilton's openness and generosity--it really attracted me. I can't
say that most mathematicians act like that.

"I was working on different things, though occasionally I would think
about the Ricci flow," Perelman added. "You didn't have to be a great
mathematician to see that this would be useful for geometrization. I
felt I didn't know very much. I kept asking questions."

Shing-Tung Yau was also asking Hamilton questions about the Ricci flow.
Yau and Hamilton had met in the seventies, and had become close,
despite considerable differences in temperament and background. A
mathematician at the University of California at San Diego who knows
both men called them "the mathematical loves of each other's lives."

Yau's family moved to Hong Kong from mainland China in 1949, when he
was five months old, along with hundreds of thousands of other refugees
fleeing Mao's armies. The previous year, his father, a relief worker
for the United Nations, had lost most of the family's savings in a
series of failed ventures. In Hong Kong, to support his wife and eight
children, he tutored college students in classical Chinese literature
and philosophy.

When Yau was fourteen, his father died of kidney cancer, leaving his
mother dependent on handouts from Christian missionaries and whatever
small sums she earned from selling handicrafts. Until then, Yau had
been an indifferent student. But he began to devote himself to
schoolwork, tutoring other students in math to make money. "Part of the
thing that drives Yau is that he sees his own life as being his
father's revenge," said Dan Stroock, the M.I.T. mathematician, who has
known Yau for twenty years. "Yau's father was like the Talmudist whose
children are starving."

Yau studied math at the Chinese University of Hong Kong, where he
attracted the attention of Shiing-Shen Chern, the preeminent Chinese
mathematician, who helped him win a scholarship to the University of
California at Berkeley. Chern was the author of a famous theorem
combining topology and geometry. He spent most of his career in the
United States, at Berkeley. He made frequent visits to Hong Kong,
Taiwan, and, later, China, where he was a revered symbol of Chinese
intellectual achievement, to promote the study of math and science.

In 1969, Yau started graduate school at Berkeley, enrolling in seven
graduate courses each term and auditing several others. He sent half of
his scholarship money back to his mother in China and impressed his
professors with his tenacity. He was obliged to share credit for his
first major result when he learned that two other mathematicians were
working on the same problem. In 1976, he proved a twenty-year-old
conjecture pertaining to a type of manifold that is now crucial to
string theory. A French mathematician had formulated a proof of the
problem, which is known as Calabi's conjecture, but Yau's, because it
was more general, was more powerful. (Physicists now refer to
Calabi-Yau manifolds.) "He was not so much thinking up some original
way of looking at a subject but solving extremely hard technical
problems that at the time only he could solve, by sheer intellect and
force of will," Phillip Griffiths, a geometer and a former director of
the Institute for Advanced Study, said.

In 1980, when Yau was thirty, he became one of the youngest
mathematicians ever to be appointed to the permanent faculty of the
Institute for Advanced Study, and he began to attract talented
students. He won a Fields Medal two years later, the first Chinese ever
to do so. By this time, Chern was seventy years old and on the verge of
retirement. According to a relative of Chern's, "Yau decided that he
was going to be the next famous Chinese mathematician and that it was
time for Chern to step down."

Harvard had been trying to recruit Yau, and when, in 1983, it was about
to make him a second offer Phillip Griffiths told the dean of faculty a
version of a story from "The Romance of the Three Kingdoms," a Chinese
classic. In the third century A.D., a Chinese warlord dreamed of
creating an empire, but the most brilliant general in China was working
for a rival. Three times, the warlord went to his enemy's kingdom to
seek out the general. Impressed, the general agreed to join him, and
together they succeeded in founding a dynasty. Taking the hint, the
dean flew to Philadelphia, where Yau lived at the time, to make him an
offer. Even so, Yau turned down the job. Finally, in 1987, he agreed to
go to Harvard.

Yau's entrepreneurial drive extended to collaborations with colleagues
and students, and, in addition to conducting his own research, he began
organizing seminars. He frequently allied himself with brilliantly
inventive mathematicians, including Richard Schoen and William Meeks.
But Yau was especially impressed by Hamilton, as much for his swagger
as for his imagination. "I can have fun with Hamilton," Yau told us
during the string-theory conference in Beijing. "I can go swimming with
him. I go out with him and his girlfriends and all that." Yau was
convinced that Hamilton could use the Ricci-flow equation to solve the
Poincaré and Thurston conjectures, and he urged him to focus on the
problems. "Meeting Yau changed his mathematical life," a friend of both
mathematicians said of Hamilton. "This was the first time he had been
on to something extremely big. Talking to Yau gave him courage and
direction."

Yau believed that if he could help solve the Poincaré it would be a
victory not just for him but also for China. In the mid-nineties, Yau
and several other Chinese scholars began meeting with President Jiang
Zemin to discuss how to rebuild the country's scientific institutions,
which had been largely destroyed during the Cultural Revolution.
Chinese universities were in dire condition. According to Steve Smale,
who won a Fields for proving the Poincaré in higher dimensions, and
who, after retiring from Berkeley, taught in Hong Kong, Peking
University had "halls filled with the smell of urine, one common room,
one office for all the assistant professors," and paid its faculty
wretchedly low salaries. Yau persuaded a Hong Kong real-estate mogul to
help finance a mathematics institute at the Chinese Academy of
Sciences, in Beijing, and to endow a Fields-style medal for Chinese
mathematicians under the age of forty-five. On his trips to China, Yau
touted Hamilton and their joint work on the Ricci flow and the Poincaré
as a model for young Chinese mathematicians. As he put it in Beijing,
"They always say that the whole country should learn from Mao or some
big heroes. So I made a joke to them, but I was half serious. I said
the whole country should learn from Hamilton."

Grigory Perelman was learning from Hamilton already. In 1993, he began
a two-year fellowship at Berkeley. While he was there, Hamilton gave
several talks on campus, and in one he mentioned that he was working on
the Poincaré. Hamilton's Ricci-flow strategy was extremely technical
and tricky to execute. After one of his talks at Berkeley, he told
Perelman about his biggest obstacle. As a space is smoothed under the
Ricci flow, some regions deform into what mathematicians refer to as
"singularities." Some regions, called "necks," become attenuated areas
of infinite density. More troubling to Hamilton was a kind of
singularity he called the "cigar." If cigars formed, Hamilton worried,
it might be impossible to achieve uniform geometry. Perelman realized
that a paper he had written on Alexandrov spaces might help Hamilton
prove Thurston's conjecture--and the Poincaré--once Hamilton solved the
cigar problem. "At some point, I asked Hamilton if he knew a certain
collapsing result that I had proved but not published--which turned out
to be very useful," Perelman said. "Later, I realized that he didn't
understand what I was talking about." Dan Stroock, of M.I.T., said,
"Perelman may have learned stuff from Yau and Hamilton, but, at the
time, they were not learning from him."

By the end of his first year at Berkeley, Perelman had written several
strikingly original papers. He was asked to give a lecture at the 1994
I.M.U. congress, in Zurich, and invited to apply for jobs at Stanford,
Princeton, the Institute for Advanced Study, and the University of Tel
Aviv. Like Yau, Perelman was a formidable problem solver. Instead of
spending years constructing an intricate theoretical framework, or
defining new areas of research, he focussed on obtaining particular
results. According to Mikhail Gromov, a renowned Russian geometer who
has collaborated with Perelman, he had been trying to overcome a
technical difficulty relating to Alexandrov spaces and had apparently
been stumped. "He couldn't do it," Gromov said. "It was hopeless."

Perelman told us that he liked to work on several problems at once. At
Berkeley, however, he found himself returning again and again to
Hamilton's Ricci-flow equation and the problem that Hamilton thought he
could solve with it. Some of Perelman's friends noticed that he was
becoming more and more ascetic. Visitors from St. Petersburg who stayed
in his apartment were struck by how sparsely furnished it was. Others
worried that he seemed to want to reduce life to a set of rigid axioms.
When a member of a hiring committee at Stanford asked him for a C.V. to
include with requests for letters of recommendation, Perelman balked.
"If they know my work, they don't need my C.V.," he said. "If they need
my C.V., they don't know my work."

Ultimately, he received several job offers. But he declined them all,
and in the summer of 1995 returned to St. Petersburg, to his old job at
the Steklov Institute, where he was paid less than a hundred dollars a
month. (He told a friend that he had saved enough money in the United
States to live on for the rest of his life.) His father had moved to
Israel two years earlier, and his younger sister was planning to join
him there after she finished college. His mother, however, had decided
to remain in St. Petersburg, and Perelman moved in with her. "I realize
that in Russia I work better," he told colleagues at the Steklov.

At twenty-nine, Perelman was firmly established as a mathematician and
yet largely unburdened by professional responsibilities. He was free to
pursue whatever problems he wanted to, and he knew that his work,
should he choose to publish it, would be shown serious consideration.
Yakov Eliashberg, a mathematician at Stanford who knew Perelman at
Berkeley, thinks that Perelman returned to Russia in order to work on
the Poincaré. "Why not?" Perelman said when we asked whether
Eliashberg's hunch was correct.

The Internet made it possible for Perelman to work alone while
continuing to tap a common pool of knowledge. Perelman searched
Hamilton's papers for clues to his thinking and gave several seminars
on his work. "He didn't need any help," Gromov said. "He likes to be
alone. He reminds me of Newton--this obsession with an idea, working by
yourself, the disregard for other people's opinion. Newton was more
obnoxious. Perelman is nicer, but very obsessed."

In 1995, Hamilton published a paper in which he discussed a few of his
ideas for completing a proof of the Poincaré. Reading the paper,
Perelman realized that Hamilton had made no progress on overcoming his
obstacles--the necks and the cigars. "I hadn't seen any evidence of
progress after early 1992," Perelman told us. "Maybe he got stuck even
earlier." However, Perelman thought he saw a way around the impasse. In
1996, he wrote Hamilton a long letter outlining his notion, in the hope
of collaborating. "He did not answer," Perelman said. "So I decided to
work alone."

Yau had no idea that Hamilton's work on the Poincaré had stalled. He
was increasingly anxious about his own standing in the mathematics
profession, particularly in China, where, he worried, a younger scholar
could try to supplant him as Chern's heir. More than a decade had
passed since Yau had proved his last major result, though he continued
to publish prolifically. "Yau wants to be the king of geometry,"
Michael Anderson, a geometer at Stony Brook, said. "He believes that
everything should issue from him, that he should have oversight. He
doesn't like people encroaching on his territory." Determined to retain
control over his field, Yau pushed his students to tackle big problems.
At Harvard, he ran a notoriously tough seminar on differential
geometry, which met for three hours at a time three times a week. Each
student was assigned a recently published proof and asked to
reconstruct it, fixing any errors and filling in gaps. Yau believed
that a mathematician has an obligation to be explicit, and impressed on
his students the importance of step-by-step rigor.

There are two ways to get credit for an original contribution in
mathematics. The first is to produce an original proof. The second is
to identify a significant gap in someone else's proof and supply the
missing chunk. However, only true mathematical gaps--missing or
mistaken arguments--can be the basis for a claim of originality.
Filling in gaps in exposition--shortcuts and abbreviations used to make
a proof more efficient--does not count. When, in 1993, Andrew Wiles
revealed that a gap had been found in his proof of Fermat's last
theorem, the problem became fair game for anyone, until, the following
year, Wiles fixed the error. Most mathematicians would agree that, by
contrast, if a proof's implicit steps can be made explicit by an
expert, then the gap is merely one of exposition, and the proof should
be considered complete and correct.

Occasionally, the difference between a mathematical gap and a gap in
exposition can be hard to discern. On at least one occasion, Yau and
his students have seemed to confuse the two, making claims of
originality that other mathematicians believe are unwarranted. In 1996,
a young geometer at Berkeley named Alexander Givental had proved a
mathematical conjecture about mirror symmetry, a concept that is
fundamental to string theory. Though other mathematicians found
Givental's proof hard to follow, they were optimistic that he had
solved the problem. As one geometer put it, "Nobody at the time said it
was incomplete and incorrect."

In the fall of 1997, Kefeng Liu, a former student of Yau's who taught
at Stanford, gave a talk at Harvard on mirror symmetry. According to
two geometers in the audience, Liu proceeded to present a proof
strikingly similar to Givental's, describing it as a paper that he had
co-authored with Yau and another student of Yau's. "Liu mentioned
Givental but only as one of a long list of people who had contributed
to the field," one of the geometers said. (Liu maintains that his proof
was significantly different from Givental's.)

Around the same time, Givental received an e-mail signed by Yau and his
collaborators, explaining that they had found his arguments impossible
to follow and his notation baffling, and had come up with a proof of
their own. They praised Givental for his "brilliant idea" and wrote,
"In the final version of our paper your important contribution will be
acknowledged."

A few weeks later, the paper, "Mirror Principle I," appeared in the
Asian Journal of Mathematics, which is co-edited by Yau. In it, Yau and
his coauthors describe their result as "the first complete proof" of
the mirror conjecture. They mention Givental's work only in passing.
"Unfortunately," they write, his proof, "which has been read by many
prominent experts, is incomplete." However, they did not identify a
specific mathematical gap.

Givental was taken aback. "I wanted to know what their objection was,"
he told us. "Not to expose them or defend myself." In March, 1998, he
published a paper that included a three-page footnote in which he
pointed out a number of similarities between Yau's proof and his own.
Several months later, a young mathematician at the University of
Chicago who was asked by senior colleagues to investigate the dispute
concluded that Givental's proof was complete. Yau says that he had been
working on the proof for years with his students and that they achieved
their result independently of Givental. "We had our own ideas, and we
wrote them up," he says.

Around this time, Yau had his first serious conflict with Chern and the
Chinese mathematical establishment. For years, Chern had been hoping to
bring the I.M.U.'s congress to Beijing. According to several
mathematicians who were active in the I.M.U. at the time, Yau made an
eleventh-hour effort to have the congress take place in Hong Kong
instead. But he failed to persuade a sufficient number of colleagues to
go along with his proposal, and the I.M.U. ultimately decided to hold
the 2002 congress in Beijing. (Yau denies that he tried to bring the
congress to Hong Kong.) Among the delegates the I.M.U. appointed to a
group that would be choosing speakers for the congress was Yau's most
successful student, Gang Tian, who had been at N.Y.U. with Perelman and
was now a professor at M.I.T. The host committee in Beijing also asked
Tian to give a plenary address.

Yau was caught by surprise. In March, 2000, he had published a survey
of recent research in his field studded with glowing references to Tian
and to their joint projects. He retaliated by organizing his first
conference on string theory, which opened in Beijing a few days before
the math congress began, in late August, 2002. He persuaded Stephen
Hawking and several Nobel laureates to attend, and for days the Chinese
newspapers were full of pictures of famous scientists. Yau even managed
to arrange for his group to have an audience with Jiang Zemin. A
mathematician who helped organize the math congress recalls that along
the highway between Beijing and the airport there were "billboards with
pictures of Stephen Hawking plastered everywhere."

That summer, Yau wasn't thinking much about the Poincaré. He had
confidence in Hamilton, despite his slow pace. "Hamilton is a very good
friend," Yau told us in Beijing. "He is more than a friend. He is a
hero. He is so original. We were working to finish our proof. Hamilton
worked on it for twenty-five years. You work, you get tired. He
probably got a little tired--and you want to take a rest."

Then, on November 12, 2002, Yau received an e-mail message from a
Russian mathematician whose name didn't immediately register. "May I
bring to your attention my paper," the e-mail said.

On November 11th, Perelman had posted a thirty-nine-page paper entitled
"The Entropy Formula for the Ricci Flow and Its Geometric
Applications," on arXiv.org, a Web site used by mathematicians to post
preprints--articles awaiting publication in refereed journals. He then
e-mailed an abstract of his paper to a dozen mathematicians in the
United States--including Hamilton, Tian, and Yau--none of whom had
heard from him for years. In the abstract, he explained that he had
written "a sketch of an eclectic proof" of the geometrization
conjecture.

Perelman had not mentioned the proof or shown it to anyone. "I didn't
have any friends with whom I could discuss this," he said in St.
Petersburg. "I didn't want to discuss my work with someone I didn't
trust." Andrew Wiles had also kept the fact that he was working on
Fermat's last theorem a secret, but he had had a colleague vet the
proof before making it public. Perelman, by casually posting a proof on
the Internet of one of the most famous problems in mathematics, was not
just flouting academic convention but taking a considerable risk. If
the proof was flawed, he would be publicly humiliated, and there would
be no way to prevent another mathematician from fixing any errors and
claiming victory. But Perelman said he was not particularly concerned.
"My reasoning was: if I made an error and someone used my work to
construct a correct proof I would be pleased," he said. "I never set
out to be the sole solver of the Poincaré."

Gang Tian was in his office at M.I.T. when he received Perelman's
e-mail. He and Perelman had been friendly in 1992, when they were both
at N.Y.U. and had attended the same weekly math seminar in Princeton.
"I immediately realized its importance," Tian said of Perelman's paper.
Tian began to read the paper and discuss it with colleagues, who were
equally enthusiastic.

On November 19th, Vitali Kapovitch, a geometer, sent Perelman an e-mail:

Hi Grisha, Sorry to bother you but a lot of people are asking me about
your preprint "The entropy formula for the Ricci..." Do I understand
it correctly that while you cannot yet do all the steps in the Hamilton
program you can do enough so that using some collapsing results you can
prove geometrization? Vitali.

Perelman's response, the next day, was terse: "That's correct. Grisha."

In fact, what Perelman had posted on the Internet was only the first
installment of his proof. But it was sufficient for mathematicians to
see that he had figured out how to solve the Poincaré. Barry Mazur, the
Harvard mathematician, uses the image of a dented fender to describe
Perelman's achievement: "Suppose your car has a dented fender and you
call a mechanic to ask how to smooth it out. The mechanic would have a
hard time telling you what to do over the phone. You would have to
bring the car into the garage for him to examine. Then he could tell
you where to give it a few knocks. What Hamilton introduced and
Perelman completed is a procedure that is independent of the
particularities of the blemish. If you apply the Ricci flow to a 3-D
space, it will begin to undent it and smooth it out. The mechanic would
not need to even see the car--just apply the equation." Perelman proved
that the "cigars" that had troubled Hamilton could not actually occur,
and he showed that the "neck" problem could be solved by performing an
intricate sequence of mathematical surgeries: cutting out singularities
and patching up the raw edges. "Now we have a procedure to smooth
things and, at crucial points, control the breaks," Mazur said.

Tian wrote to Perelman, asking him to lecture on his paper at M.I.T.
Colleagues at Princeton and Stony Brook extended similar invitations.
Perelman accepted them all and was booked for a month of lectures
beginning in April, 2003. "Why not?" he told us with a shrug. Speaking
of mathematicians generally, Fedor Nazarov, a mathematician at Michigan
State University, said, "After you've solved a problem, you have a
great urge to talk about it."

Hamilton and Yau were stunned by Perelman's announcement. "We felt that
nobody else would be able to discover the solution," Yau told us in
Beijing. "But then, in 2002, Perelman said that he published something.
He basically did a shortcut without doing all the detailed estimates
that we did." Moreover, Yau complained, Perelman's proof "was written
in such a messy way that we didn't understand."

Perelman's April lecture tour was treated by mathematicians and by the
press as a major event. Among the audience at his talk at Princeton
were John Ball, Andrew Wiles, John Forbes Nash, Jr., who had proved the
Riemannian embedding theorem, and John Conway, the inventor of the
cellular automaton game Life. To the astonishment of many in the
audience, Perelman said nothing about the Poincaré. "Here is a guy who
proved a world-famous theorem and didn't even mention it," Frank Quinn,
a mathematician at Virginia Tech, said. "He stated some key points and
special properties, and then answered questions. He was establishing
credibility. If he had beaten his chest and said, 'I solved it,' he
would have got a huge amount of resistance." He added, "People were
expecting a strange sight. Perelman was much more normal than they
expected."

To Perelman's disappointment, Hamilton did not attend that lecture or
the next ones, at Stony Brook. "I'm a disciple of Hamilton's, though I
haven't received his authorization," Perelman told us. But John Morgan,
at Columbia, where Hamilton now taught, was in the audience at Stony
Brook, and after a lecture he invited Perelman to speak at Columbia.
Perelman, hoping to see Hamilton, agreed. The lecture took place on a
Saturday morning. Hamilton showed up late and asked no questions during
either the long discussion session that followed the talk or the lunch
after that. "I had the impression he had read only the first part of my
paper," Perelman said.

In the April 18, 2003, issue of Science, Yau was featured in an article
about Perelman's proof: "Many experts, although not all, seem convinced
that Perelman has stubbed out the cigars and tamed the narrow necks.
But they are less confident that he can control the number of
surgeries. That could prove a fatal flaw, Yau warns, noting that many
other attempted proofs of the Poincaré conjecture have stumbled over
similar missing steps." Proofs should be treated with skepticism until
mathematicians have had a chance to review them thoroughly, Yau told
us. Until then, he said, "it's not math--it's religion."

By mid-July, Perelman had posted the final two installments of his
proof on the Internet, and mathematicians had begun the work of formal
explication, painstakingly retracing his steps. In the United States,
at least two teams of experts had assigned themselves this task: Gang
Tian (Yau's rival) and John Morgan; and a pair of researchers at the
University of Michigan. Both projects were supported by the Clay
Institute, which planned to publish Tian and Morgan's work as a book.
The book, in addition to providing other mathematicians with a guide to
Perelman's logic, would allow him to be considered for the Clay
Institute's million-dollar prize for solving the Poincaré. (To be
eligible, a proof must be published in a peer-reviewed venue and
withstand two years of scrutiny by the mathematical community.)

On September 10, 2004, more than a year after Perelman returned to St.
Petersburg, he received a long e-mail from Tian, who said that he had
just attended a two-week workshop at Princeton devoted to Perelman's
proof. "I think that we have understood the whole paper," Tian wrote.
"It is all right."

Perelman did not write back. As he explained to us, "I didn't worry too
much myself. This was a famous problem. Some people needed time to get
accustomed to the fact that this is no longer a conjecture. I
personally decided for myself that it was right for me to stay away
from verification and not to participate in all these meetings. It is
important for me that I don't influence this process."

In July of that year, the National Science Foundation had given nearly
a million dollars in grants to Yau, Hamilton, and several students of
Yau's to study and apply Perelman's "breakthrough." An entire branch of
mathematics had grown up around efforts to solve the Poincaré, and now
that branch appeared at risk of becoming obsolete. Michael Freedman,
who won a Fields for proving the Poincaré conjecture for the fourth
dimension, told the Times that Perelman's proof was a "small sorrow for
this particular branch of topology." Yuri Burago said, "It kills the
field. After this is done, many mathematicians will move to other
branches of mathematics."

Five months later, Chern died, and Yau's efforts to insure that
he---not Tian--was recognized as his successor turned vicious. "It's
all about their primacy in China and their leadership among the
expatriate Chinese," Joseph Kohn, a former chairman of the Prince-ton
mathematics department, said. "Yau's not jealous of Tian's mathematics,
but he's jealous of his power back in China."

Though Yau had not spent more than a few months at a time on mainland
China since he was an infant, he was convinced that his status as the
only Chinese Fields Medal winner should make him Chern's successor. In
a speech he gave at Zhejiang University, in Hangzhou, during the summer
of 2004, Yau reminded his listeners of his Chinese roots. "When I
stepped out from the airplane, I touched the soil of Beijing and felt
great joy to be in my mother country," he said. "I am proud to say that
when I was awarded the Fields Medal in mathematics, I held no passport
of any country and should certainly be considered Chinese."

The following summer, Yau returned to China and, in a series of
interviews with Chinese reporters, attacked Tian and the mathematicians
at Peking University. In an article published in a Beijing science
newspaper, which ran under the headline "SHING-TUNG YAU IS SLAMMING
ACADEMIC CORRUPTION IN CHINA," Yau called Tian "a complete mess." He
accused him of holding multiple professorships and of collecting a
hundred and twenty-five thousand dollars for a few months' work at a
Chinese university, while students were living on a hundred dollars a
month. He also charged Tian with shoddy scholarship and plagiarism, and
with intimidating his graduate students into letting him add his name
to their papers. "Since I promoted him all the way to his academic fame
today, I should also take responsibility for his improper behavior,"
Yau was quoted as saying to a reporter, explaining why he felt obliged
to speak out.

In another interview, Yau described how the Fields committee had passed
Tian over in 1988 and how he had lobbied on Tian's behalf with various
prize committees, including one at the National Science Foundation,
which awarded Tian five hundred thousand dollars in 1994.

Tian was appalled by Yau's attacks, but he felt that, as Yau's former
student, there was little he could do about them. "His accusations were
baseless," Tian told us. But, he added, "I have deep roots in Chinese
culture. A teacher is a teacher. There is respect. It is very hard for
me to think of anything to do."

While Yau was in China, he visited Xi-Ping Zhu, a protégé of his who
was now chairman of the mathematics department at Sun Yat-sen
University. In the spring of 2003, after Perelman completed his lecture
tour in the United States, Yau had recruited Zhu and another student,
Huai-Dong Cao, a professor at Lehigh University, to undertake an
explication of Perelman's proof. Zhu and Cao had studied the Ricci flow
under Yau, who considered Zhu, in particular, to be a mathematician of
exceptional promise. "We have to figure out whether Perelman's paper
holds together," Yau told them. Yau arranged for Zhu to spend the
2005-06 academic year at Harvard, where he gave a seminar on Perelman's
proof and continued to work on his paper with Cao.

On April 13th of this year, the thirty-one mathematicians on the
editorial board of the Asian Journal of Mathematics received a brief
e-mail from Yau and the journal's co-editor informing them that they
had three days to comment on a paper by Xi-Ping Zhu and Huai-Dong Cao
titled "The Hamilton-Perelman Theory of Ricci Flow: The Poincaré and
Geometrization Conjectures," which Yau planned to publish in the
journal. The e-mail did not include a copy of the paper, reports from
referees, or an abstract. At least one board member asked to see the
paper but was told that it was not available. On April 16th, Cao
received a message from Yau telling him that the paper had been
accepted by the A.J.M., and an abstract was posted on the journal's Web
site.

A month later, Yau had lunch in Cambridge with Jim Carlson, the
president of the Clay Institute. He told Carlson that he wanted to
trade a copy of Zhu and Cao's paper for a copy of Tian and Morgan's
book manuscript. Yau told us he was worried that Tian would try to
steal from Zhu and Cao's work, and he wanted to give each party
simultaneous access to what the other had written. "I had a lunch with
Carlson to request to exchange both manuscripts to make sure that
nobody can copy the other," Yau said. Carlson demurred, explaining that
the Clay Institute had not yet received Tian and Morgan's complete
manuscript.

By the end of the following week, the title of Zhu and Cao's paper on
the A.J.M.'s Web site had changed, to "A Complete Proof of the Poincaré
and Geometrization Conjectures: Application of the Hamilton-Perelman
Theory of the Ricci Flow." The abstract had also been revised. A new
sentence explained, "This proof should be considered as the crowning
achievement of the Hamilton-Perelman theory of Ricci flow."

Zhu and Cao's paper was more than three hundred pages long and filled
the A.J.M.'s entire June issue. The bulk of the paper is devoted to
reconstructing many of Hamilton's Ricci-flow results--including results
that Perelman had made use of in his proof--and much of Perelman's
proof of the Poincaré. In their introduction, Zhu and Cao credit
Perelman with having "brought in fresh new ideas to figure out
important steps to overcome the main obstacles that remained in the
program of Hamilton." However, they write, they were obliged to
"substitute several key arguments of Perelman by new approaches based
on our study, because we were unable to comprehend these original
arguments of Perelman which are essential to the completion of the
geometrization program." Mathematicians familiar with Perelman's proof
disputed the idea that Zhu and Cao had contributed significant new
approaches to the Poincaré. "Perelman already did it and what he did
was complete and correct," John Morgan said. "I don't see that they did
anything different."

By early June, Yau had begun to promote the proof publicly. On June
3rd, at his mathematics institute in Beijing, he held a press
conference. The acting director of the mathematics institute,
attempting to explain the relative contributions of the different
mathematicians who had worked on the Poincaré, said, "Hamilton
contributed over fifty per cent; the Russian, Perelman, about
twenty-five per cent; and the Chinese, Yau, Zhu, and Cao et al., about
thirty per cent." (Evidently, simple addition can sometimes trip up
even a mathematician.) Yau added, "Given the significance of the
Poincaré, that Chinese mathematicians played a thirty-per-cent role is
by no means easy. It is a very important contribution."

On June 12th, the week before Yau's conference on string theory opened
in Beijing, the South China Morning Post reported, "Mainland
mathematicians who helped crack a 'millennium math problem' will
present the methodology and findings to physicist Stephen Hawking...
Yau Shing-Tung, who organized Professor Hawking's visit and is also
Professor Cao's teacher, said yesterday he would present the findings
to Professor Hawking because he believed the knowledge would help his
research into the formation of black holes."

On the morning of his lecture in Beijing, Yau told us, "We want our
contribution understood. And this is also a strategy to encourage Zhu,
who is in China and who has done really spectacular work. I mean,
important work with a century-long problem, which will probably have
another few century-long implications. If you can attach your name in
any way, it is a contribution."

E. T. Bell, the author of "Men of Mathematics," a witty history of the
discipline published in 1937, once lamented "the squabbles over
priority which disfigure scientific history." But in the days before
e-mail, blogs, and Web sites, a certain decorum usually prevailed. In
1881, Poincaré, who was then at the University of Caen, had an
altercation with a German mathematician in Leipzig named Felix Klein.
Poincaré had published several papers in which he labelled certain
functions "Fuchsian," after another mathematician. Klein wrote to
Poincaré, pointing out that he and others had done significant work on
these functions, too. An exchange of polite letters between Leipzig and
Caen ensued. Poincaré's last word on the subject was a quote from
Goethe's "Faust": "Name ist Schall und Rauch." Loosely translated, that
corresponds to Shakespeare's "What's in a name?"

This, essentially, is what Yau's friends are asking themselves. "I find
myself getting annoyed with Yau that he seems to feel the need for more
kudos," Dan Stroock, of M.I.T., said. "This is a guy who did
magnificent things, for which he was magnificently rewarded. He won
every prize to be won. I find it a little mean of him to seem to be
trying to get a share of this as well." Stroock pointed out that,
twenty-five years ago, Yau was in a situation very similar to the one
Perelman is in today. His most famous result, on Calabi-Yau manifolds,
was hugely important for theoretical physics. "Calabi outlined a
program," Stroock said. "In a real sense, Yau was Calabi's Perelman.
Now he's on the other side. He's had no compunction at all in taking
the lion's share of credit for Calabi-Yau. And now he seems to be
resenting Perelman getting credit for completing Hamilton's program. I
don't know if the analogy has ever occurred to him."

Mathematics, more than many other fields, depends on collaboration.
Most problems require the insights of several mathematicians in order
to be solved, and the profession has evolved a standard for crediting
individual contributions that is as stringent as the rules governing
math itself. As Perelman put it, "If everyone is honest, it is natural
to share ideas." Many mathematicians view Yau's conduct over the
Poincaré as a violation of this basic ethic, and worry about the damage
it has caused the profession. "Politics, power, and control have no
legitimate role in our community, and they threaten the integrity of
our field," Phillip Griffiths said.

Perelman likes to attend opera performances at the Mariinsky Theatre,
in St. Petersburg. Sitting high up in the back of the house, he can't
make out the singers' expressions or see the details of their costumes.
But he cares only about the sound of their voices, and he says that the
acoustics are better where he sits than anywhere else in the theatre.
Perelman views the mathematics community--and much of the larger
world--from a similar remove.

Before we arrived in St. Petersburg, on June 23rd, we had sent several
messages to his e-mail address at the Steklov Institute, hoping to
arrange a meeting, but he had not replied. We took a taxi to his
apartment building and, reluctant to intrude on his privacy, left a
book--a collection of John Nash's papers--in his mailbox, along with a
card saying that we would be sitting on a bench in a nearby playground
the following afternoon. The next day, after Perelman failed to appear,
we left a box of pearl tea and a note describing some of the questions
we hoped to discuss with him. We repeated this ritual a third time.
Finally, believing that Perelman was out of town, we pressed the buzzer
for his apartment, hoping at least to speak with his mother. A woman
answered and let us inside. Perelman met us in the dimly lit hallway of
the apartment. It turned out that he had not checked his Steklov e-mail
address for months, and had not looked in his mailbox all week. He had
no idea who we were.

We arranged to meet at ten the following morning on Nevsky Prospekt.
From there, Perelman, dressed in a sports coat and loafers, took us on
a four-hour walking tour of the city, commenting on every building and
vista. After that, we all went to a vocal competition at the St.
Petersburg Conservatory, which lasted for five hours. Perelman
repeatedly said that he had retired from the mathematics community and
no longer considered himself a professional mathematician. He mentioned
a dispute that he had had years earlier with a collaborator over how to
credit the author of a particular proof, and said that he was dismayed
by the discipline's lax ethics. "It is not people who break ethical
standards who are regarded as aliens," he said. "It is people like me
who are isolated." We asked him whether he had read Cao and Zhu's
paper. "It is not clear to me what new contribution did they make," he
said. "Apparently, Zhu did not quite understand the argument and
reworked it." As for Yau, Perelman said, "I can't say I'm outraged.
Other people do worse. Of course, there are many mathematicians who are
more or less honest. But almost all of them are conformists. They are
more or less honest, but they tolerate those who are not honest."

The prospect of being awarded a Fields Medal had forced him to make a
complete break with his profession. "As long as I was not conspicuous,
I had a choice," Perelman explained. "Either to make some ugly
thing"--a fuss about the math community's lack of integrity--"or, if I
didn't do this kind of thing, to be treated as a pet. Now, when I
become a very conspicuous person, I cannot stay a pet and say nothing.
That is why I had to quit." We asked Perelman whether, by refusing the
Fields and withdrawing from his profession, he was eliminating any
possibility of influencing the discipline. "I am not a politician!" he
replied, angrily. Perelman would not say whether his objection to
awards extended to the Clay Institute's million-dollar prize. "I'm not
going to decide whether to accept the prize until it is offered," he
said.

Mikhail Gromov, the Russian geometer, said that he understood Perelman's logic: "To do great work, you have to have a pure mind. You can think only about the mathematics. Everything else is human weakness. Accepting prizes is showing weakness." Others might view Perelman's refusal to accept a Fields as arrogant, Gromov said, but his principles are admirable. "The ideal scientist does science and cares about nothing else," he said. "He wants to live this ideal. Now, I don't think he really lives on this ideal plane. But he wants to."