MANIFOLD
DESTINY
by SYLVIA NASAR AND DAVID GRUBER
From the New Yorker magazine
Issue of 2006-08-28 and posted online
on 2006-08-21
A legendary problem and the battle over
who solved it.
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 preëminent
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.”
(To Page Top) |
The Emperor of Math
By DENNIS OVERBYE
New York Times
October 17, 2006
In 1979, Shing-Tung Yau, then a mathematician at the Institute
for Advanced Study in Princeton,
was visiting China and
asked the authorities for permission to visit his birthplace,
Shantou, a mountain town in Guangdong Province.
At first they refused, saying the town was not on the map.
Finally, after more delays and excuses, Dr. Yau found himself
being driven on a fresh dirt road through farm fields to his
hometown, where the citizens slaughtered a cow to celebrate
his homecoming. Only long after he left did Dr. Yau learn that
the road had been built for his visit.
“I was truly amazed,” Dr. Yau said recently, smiling
sheepishly. “I feel guilty that this happened.” He
was standing in the airy frosted-glass light of his office
in the Morningside Center of Mathematics, one of three math
institutes he has founded in China.
For nine months of the year, Dr. Yau is a Harvard math
professor, best known for inventing the mathematical structures
known as Calabi-Yau spaces that underlie string theory, the
supposed
“theory of everything.” In 1982 he won a Fields Medal, the mathematics
equivalent of a Nobel
Prize. Dr. Yau can be found holding court in the Yenching restaurant in
Harvard Square or off the math library in his cramped office, where the blackboard
is covered with equations and sketches of artfully chopped-up doughnuts.
But the other three months he is what his friend Andrew Strominger,
a Harvard physicist, called “the emperor ascendant of
Chinese science,”
one of the most prominent of the “overseas Chinese” who
return home every summer to work, teach, lobby, inspire and
feud like warlords in an effort to advance world-class science
in China.
David J. Gross, the Nobel physicist and string theorist who
directs the Kavli Institute for Theoretical Physics in Santa
Barbara, called Dr. Yau “a transitional figure, between
emperor and democrat.”
Dr. Yau’s story is a window into the dynamics that prevail
in China as 5,000 years of Middle Kingdom tradition tries to
mix with postmodern science, a blending that, if it takes,
could eventually reshape the balance of science and technology
in the world.
“In China he is a movie star,” said Ronnie Chan,
a Hong Kong real estate developer and an old friend who helped
bankroll the Morningside Center. And last summer Dr. Yau played
the part, dashing in black cars from television studios to
V.I.P. receptions in forbidden gardens in the Forbidden City.
He ushered Stephen Hawking into the Great Hall of the People
in Tiananmen Square to kick off a meeting of some of the world’s
leading physicists on string theory, and beamed as a poem he
had written was performed by a music professor on the conference
stage. It reads in part: “Beautiful indeed/is the source
of truth./To measure the changes of time and space/the smartest
are nothing.”
Dr. Yau does not buy the emperor bit. Where, he protested
recently, is his empire if he holds no political position and
two of his most brilliant recent students are currently without
jobs? “It’s just a perception as far as I can tell,” he
said.
Certainly, his life is not all roses. In the last year alone
Dr. Yau has been engaged in a very public fight with Beijing
University, having accused it of corruption, and a New Yorker
magazine article portrayed him as trying to horn in on credit
for solving the Poincaré conjecture, a famous 100-year-old
problem about the structure of space.
Everybody agrees that Dr. Yau is one of the great mathematicians
of the age.
“Yau really is a genius,” said Robert Greene,
a mathematician at the University
of California, Los Angeles. “The quantity and quality
of the math he has done is overpowering.”
But even his admirers say he has a political side. “As
Shiing-Shen Chern’s successor as emperor of Chinese mathematics,” Deane
Yang, a professor of mathematics at Polytechnic University
in Brooklyn and an old family friend, wrote in a letter to
The New Yorker, “Yau has an outsized ego and great ambition,
and has done things that dismay his peers.” But, Dr.
Yang said, Dr. Yau has been a major force for good in mathematics
and in China, a prodigious teacher who has trained 39 Ph.D.’s.
Richard Hamilton, a friend of Dr. Yau and a mathematician
at Columbia, said Dr. Yau had built “an assembly of talent,
not an empire of people, people attracted by his energy, his
brilliant ideas and his unflagging support for first-rate mathematics,
people whom Yau has brought together to work on the hardest
problems.”
A Barefoot Boy
That Shing-Tung Yau, born in 1949, had such potential was
not always obvious. His family fled the mainland and the Communist
takeover when he was a baby. As one of eight children of a
college professor and a librarian, growing up poor without
electricity or running water in a village outside Hong Kong,
he was the leader of a street gang and often skipped school.
But talks with his father instilled in him a love of literature
and philosophy and, he learned when he started studying math,
a taste for abstract thinking.
“In fact, I felt I can understand my father’s
conversations better after I learned geometry,” he said
at a talk in 2003.
When he was 14, his father died, leaving the family destitute
and in debt. To assuage his pain, the young Mr. Yau retreated
into his studies. To help out financially, he worked as a tutor.
At the Chinese University of Hong Kong, Mr. Yau emerged as
a precocious mathematician, leaving after only three years,
with no degree, for graduate school at the University of California,
Berkeley.
Mr. Yau took six courses his first semester there, leaving
scant time for lunch. By the end of his first year he had collaborated
with a teacher to prove conjectures about the geometry of unusually
warped spaces. He also came under the wing of Dr. Chern, then
widely recognized as the greatest living Chinese-born mathematician,
who told Mr. Yau he had already done enough work to write a
doctoral thesis.
Dr. Yau was in Berkeley during the wildest years of the antiwar
movement. He did not participate, but he was already political.
He and his friends demonstrated at the Taiwan Consulate General
in San Francisco to protest Japanese incursions on Chinese
territory. “Maybe we envied our American colleagues and
took after them,” Dr. Yau said.
In 1971, at age 22, Dr. Yau took his new Ph.D. to the Institute
for Advanced Study, then to the State
University of New York at Stony Brook and Stanford, where
he arrived in 1973 in time for a conference on geometry and
general relativity — Einstein’s theory that ascribes
gravity to warped space-time geometry. At the conference, Dr.
Yau had a brainstorm, realizing he could disprove a longstanding
conjecture by the University
of Pennsylvania professor Eugenio Calabi that the dimensions
of space could be curled up like the loops in a carpet.
Dr. Yau set to work on a paper. But two months later he got
a letter from Dr. Calabi and realized there was a gap in his
reasoning. “I couldn’t sleep,” Dr. Yau recalled.
After agonizing for two weeks, he concluded that the opposite
was true: the Calabi conjecture was right. His proof of that,
published in 1976, made him a star.
His paper would also lay part of the foundation 10 years later
for string theory, showing how most of the 10 dimensions of
space-time required by the “theory of everything” could
be rolled up out of sight in what are now called Calabi-Yau
spaces.
Three years later, Dr. Yau proved another important result
about Einstein’s theory of general relativity: any solution
to Einstein’s equations must have positive energy. Otherwise,
said Dr. Strominger, the Harvard physicist, space-time would
be unstable — “you could have perpetual motion.”
The result is that Dr. Yau has lived a crossover life. As
a pure mathematician, he is “a major figure, perhaps
the major figure,” as Michael Anderson of SUNY Stony
Brook called him, in building up differential geometry, the
study of curves and surfaces.
Dr. Hamilton, the Columbia mathematician, said Dr. Yau liked
to be in the center of things, unlike others who liked to retreat
into a corner and think. “He seems to thrive on being
bombarded with all this information,” he said.
He is also an honorary physicist, using “his muscular
style,” in the words of Brian Greene, a Columbia string
theorist who worked with Dr. Yau as a postdoctoral researcher
at Harvard, to smash equations and get the physics out of them. “He
corners equations like a lion after its prey,” Dr. Greene
said, “then he seals all the exits.”
Prizes and honors flowed Dr. Yau’s way after the Calabi
triumph, including the Fields Medal, a MacArthur “genius” grant
in 1985 and a National Medal of Science in 1997. He became
a United States citizen in 1990. (He said he put away the money
from the MacArthur grant for his two children’s college
education.)
A Wandering Son Returns
Dr. Yau married Yu Yun, an applied physicist from Taiwan,
in 1976. At one point, when his family had preceded him on
a move to San Diego, an institute colleague, Demetrios Christodoulou,
noticed that Dr. Yau would pick up the phone late every night
and start singing into it in Chinese.
“Yau is full of surprises, I thought to myself, now
he wants to become a great opera singer,” Dr. Christodoulou
recalled in an e-mail message. “As I later found out,
these songs were lullabies for his children.”
It was natural that as Dr. Yau’s star rose, his “mother
country,” as he put it, sought to pull him into its orbit.
When he made his first trip back to China, in 1979, Dr. Yau
became one of several returning heroes. A century of unhappy
encounters with the West had left China with a deep sense of
scientific and technological inferiority.
Dr. Yau has devoted himself to building up Chinese mathematics
and promoting basic research, arranging for Chinese students
to come to the United States, donating money and books, and
tapping rich friends to found mathematics institutes in Hong
Kong, Beijing and Hangzhou. He even lived in Taiwan in the
early 1990’s so his children would learn Chinese.
In his travels he became friendly with President Jiang
Zemin, then the leader of the Communist Party, who impressed
him as “a smart guy.” The impression was mutual.
When Mr. Jiang recited the first line of a Chinese poem at
a dinner honoring intellectuals, Dr. Yau showed off his learning
by reciting back the entire poem.
In 2004, Dr. Yau was honored at the Great Hall of the People
for his contributions to Chinese mathematics. In a speech he
said that when he won the Fields Medal, “I held no passport
of any country and should certainly be considered Chinese.”
That same year Dr. Chern died at 93. Dr. Strominger recalled
a newspaper headline declaring that with Chern’s death, “the
era of Yau”
was about to begin.
It has not been a peaceful era.
For the last year Dr. Yau has carried on a campaign against
Beijing University, accusing it of committing fraud by padding
its faculty with big names from overseas and paying them lucrative
salaries for a few months of work.
A survey in Science magazine showed that the number of such
part-time professors in China had grown to 89 from 6 over the
last six years, while the number of full-time professors had
risen to 101 from 66. The arrangement allows Chinese universities
to piggyback on the glory of work these people do in their
other jobs. Dr. Yau said it also drains resources that should
go to young researchers.
This summer, Beijing University redesignated some overseas
scholars to part time from full time. All this has taken a
toll. “Yau is not universally loved,” said Mr.
Chan, the real estate developer. “He has paid a price.”
Dr. Yau agreed. “I am completely outspoken. And I do
offend people,”
he said, adding that his style was to be intensely critical,
both of his students and of his colleagues’ ideas.
Confrontations in China go all the way to the top, because
all the money comes from the government, Dr. Yau said. “The
only reason I have the nerve to resist,” he said, “is
I’m a Harvard professor. I don’t draw a penny from
China.”
“If I didn’t have the Fields Medal,” he
added, “I would be dead to them.”
A Messy Proof
Dr. Yau’s eagerness to help China can backfire, and
that seems to have happened in the case of the Poincaré conjecture.
The conjecture, first set forth by Henri Poincaré in
1905, may be the most famous problem in mathematics and forms
part of the foundation for topology, which deals with shapes.
It says essentially that anything without holes is equivalent
to a sphere.
In 1982, Dr. Hamilton of Columbia devised a method, known
as the Ricci flow, to investigate the shapes of spaces. Dr.
Yau was enthusiastic that this method might finally crack the
Poincaré
conjecture. He began working with Dr. Hamilton and urging others
to work on it, with little success.
Then, in 2003, a Russian mathematician, Grigory Perelman,
sketched a way to jump a roadblock that had stymied Dr. Hamilton
and to prove the hallowed theorem as well as a more general
one proposed by the Cornell mathematician
William Thurston. Dr. Perelman promptly disappeared, leaving
his colleagues to connect the dots.
Among those who took up that challenge, at the urging of Dr.
Yau, were Huai-Dong Cao of Lehigh University, a former student,
and Xi-Ping Zhu of Zhongshan University. Last June, Dr. Yau
announced that they had succeeded and that the first complete
proof would appear in The Asian Journal of Mathematics, at
which he is the chief editor.
In a speech later that month during the string theory conference,
Dr. Yau said, “In Perelman’s work, many key ideas
of the proofs are sketched or outlined, but complete details
of the proofs are often missing,” adding that the Cao-Zhu
paper had filled some of these in with new arguments.
This annoyed many mathematicians, who felt that Dr. Yau had
slighted Dr. Perelman. Other teams who were finishing their
own connect-the-dots proofs said they had found no gaps in
Dr. Perelman’s work. “There was no mystery they
suddenly resolved,” said John Morgan of Columbia, who
was working with Gang Tian of Princeton on a proof.
In August, Dr. Perelman was awarded the Fields Medal at a
meeting of the International Mathematical Union in Madrid,
but he declined to accept it. A week later a drawing in The
New Yorker showed Dr. Yau trying to grab the Fields Medal from
the neck of Dr. Perelman.
On his Web site, doctoryau.com,
Dr. Yau has posted a 12-page letter showing what he and his
lawyer say are errors in the
article. The New Yorker has said it stands by its reporting. “My
name is damaged in China,” Dr. Yau said. “I have
to fix my reputation in China in order to help younger students.”
He denied that he had ever said there were gaps in Dr. Perelman’s
work. “I said it is not understood by all people,” he
said. “That is why it takes three more years.” As
a “leading geometer,” Dr. Yau said he had a duty
to dig out the truth of the proof.
Dr. Hamilton said, “In any long new work, it’s
hard to figure out what’s going on.” It was natural,
he said, that Dr. Yau would want people who had experience
in the esoteric field of Ricci flow to check the proof.
Asked if promoting the Cao-Zhu paper so loudly had been a
mistake, Dr. Yau said that even a small contribution to such
a great achievement as proving the Poincaré conjecture
would live in the history of science.
In addition, he said he wanted to encourage Dr. Zhu, who he
said had been neglected by the Chinese establishment. Dr. Yau
acknowledged that he also felt a duty to help explain Dr. Hamilton’s
work.
In a twist, a flaw has been discovered in the Cao-Zhu paper.
One of the arguments that the authors used to fill in Dr. Perelman’s
proof is identical to one posted on the Internet in June 2003
by Bruce Kleiner, of Yale,
and John Lott, of the University
of Michigan, who had been trying to explicate Dr. Perelman’s
work.
In an erratum to run in The Asian Journal of Mathematics,
Dr. Cao and Dr. Zhu acknowledge the mistake, saying they had
forgotten that they studied and incorporated that material
into their notes three years ago.
In an e-mail message, Dr. Yau said the incident was “unfortunate”
but reaffirmed his decision to expedite the paper’s publication. “Even
after the correction, the paper provides many important new
details and clarifications of Hamilton and Perelman’s
proof of the Poincaré and Thurston conjectures.”
Many mathematicians are dismayed that the Poincaré triumph
has become mired in a fight about credit and personalities. “In
spite of the rivalries,” Dr. Hamilton said, “we
are deeply dependent on each other’s work. None of us
is working in a vacuum.”
About the Poincaré proof, he said, “I’ve
never seen Yau say that Perelman hadn’t done it.” No
one, he added, had been more responsible than Dr. Yau for creating
the Ricci flow program that won Dr. Perelman his prize.
Dr. Morgan said he still regarded Dr. Yau as his friend. “He
has done tremendous things for math,” he said. “He’s
a great figure. He’s Shakespearean, larger than life.
His virtues are larger than life, and his vices are larger
than life.”
Dr. Yau said the Poincaré conjecture was bigger than
any prize and beyond politics.
“I work on mathematics because of its great beauty,” he
said. “History will judge this work, not a committee.”
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