# Grigori Perelman

**Grigori**** Yakovlevich
Perelman,** born 13 June 1966 in Leningrad, USSR
(now St. Petersburg, Russia), sometimes known as Grisha
Perelman, is a Russian
mathematician
who has made landmark contributions to Riemannian geometry and geometric topology. In particular, it appears
that he has proven Thurston's geometrization conjecture. If so, this solves in
the affirmative the famous Poincare conjecture, which has been regarded for one
hundred years as one of the most important (and most difficult) open problems
in mathematics.

In August 2006, Perelman was awarded the Fields Medal, which is widely considered
to be the top honor a mathematician can receive. However, he declined to accept
the award or appear at the congress.

## Early life and education

Grigori Perelman was born in

After graduation, Perelman began
work at the renowned Leningrad Department of Steklov Institute of Mathematics
of the USSR Academy of
Sciences in St. Petersburg,
Russia. His advisors at the Steklov
Institute were Aleksandr Danilovich
Aleksandrov and Yuri Dmitrievich
Burago. In the late 80s and early 90s, Perelman held
posts at several universities in the United States. He returned to the Steklov
Institute in 1996.

He has stated that he prefers to
stay out of the limelight, saying that "I do not think anything that I say
can be of the slightest public interest. I am not saying that because I value
my privacy, or that I am doing anything I want to
hide. There are no top-secret projects going on here. I just believe the public
has no interest in me."

## Geometrization and Poincare
conjectures

The Poincare Conjecture says "hey, you've got this alien blob that can ooze
its way out of the hold of any lasso you tie around it? Then that blob is
just an out-of-shape ball". Perelman and Hamilton proved this fact
by heating the blob up,
making it sing, stretching it like hot mozzarella and chopping it into
a million pieces. In short, the alien ain't no bagel you can swing
around with a string through his hole. (-Christina Sormani).
One of the oldest and most simply stated problems in topology is the
**Poincare Conjecture**. This conjecture states that
the only compact three dimensional simply connected manifold is a three
dimensional sphere. While most senior undergraduate math majors can
understand the statement of this conjecture the problem has baffled
mathematicians for over a century. In recent years Hamilton had been
investigating an approach to solve this problem using the Ricci Flow,
an equation which evolves and morphs a manifold into a more understandable
shape. Then in late 2002, after many years of studying Hamilton's work
and investigating the concept of entropy,
Perelman posted an article which combined with Hamilton's work would
provide a proof
of **Thurston's Geometrization Conjecture** and, thus, the Poincare Conjecture.
Since then many experts have added necessary details to Perelman's ideas,
some providing short cuts which would prove the Poincare Conjecture
directly without the difficulties involved in the complete proof of
Geometrization.

Until the autumn of 2002,
Perelman was best known for his work in comparison theorems in Riemannian geometry.
Among his notable achievements was the proof of the Soul conjecture.

### The
problem

The Poincare
conjecture, proposed by French mathematician Henri Poincare
in 1904, is the most famous open problem in topology. Loosely speaking, the conjecture
surmises that if a closed three-dimensional manifold is sufficiently like a sphere in that each loop in the manifold can be tightened to a
point, then it is really just a three-dimensional sphere. The analogous result
has been known to be true in higher dimensions for some time, however the case
of three-manifolds has turned out to be the hardest of them all, roughly
speaking because in topologically manipulating a three-manifold, there are too
few dimensions to move "problematical regions" out of the way without
interfering with something else.

In 1999, the Clay Mathematics
Institute announced the Millennium Prize
Problems ? a one million dollar prize for the proof of several
conjectures, including the Poincare conjecture. There
is universal agreement that a successful proof would constitute a landmark
event in the history of mathematics, fully comparable with the proof by Andrew Wiles of Fermat's Last Theorem,
but possibly even more far-reaching.

###
Perelman's proof

In November 2002, Perelman posted
to the ?the
first of a series of eprints in which he claimed to have
outlined a proof of the geometrization conjecture,
a result that includes the Poincare conjecture
as a particular case.

Perelman modifies Richard Hamilton's program for a proof of
the conjecture, in which the central idea is the notion of the Ricci flow.

This is similar to formulating a
dynamical process which gradually "perturbs" a given square matrix,
and which is guaranteed to result after a finite time in its rational canonical form.

Hamilton's idea had attracted a
great deal of attention, but no-one could prove that the process would not
"hang up" by developing "singularities", until Perelman's eprints
sketched a program for overcoming these obstacles. According to Perelman, a
modification of the standard Ricci flow, called Ricci flow with surgery, can
systematically excise singular regions as they develop, in a controlled way.

It is known that singularities
(including those which occur, roughly speaking, after the flow has continued
for an infinite amount of time) must occur in many cases. However,
mathematicians expect that, assuming that the geometrization
conjecture is true, any singularity which develops in a finite time is essentially
a "pinching" along certain spheres corresponding to the prime
decomposition of the 3-manifold. If so, any "infinite
time" singularities should result from certain collapsing pieces of the JSJ decomposition. Perelman's work apparently proves this claim
and thus proves the geometrization conjecture.

### Verification

Since 2003, Perelman's program
has attracted increasing attention from the mathematical community. In April 2003,
he accepted an invitation to visit Massachusetts Institute of Technology, Princeton
University, State University of New York at Stony Brook, Columbia
University and Harvard University, where
he gave a series of talks on his work.
However, after his return to

On 25 May 2006, Bruce Kleiner
and John Lott, both of the University of
Michigan, posted a paper on ?that claims to fill in the details of
Perelman's proof of the Geometrization conjecture.

In June 2006, the Asian Journal
of Mathematics published a paper by Xi-Ping Zhu of Sun Yat-sen University in China and Huai-Dong Cao of Lehigh University in Pennsylvania, claiming to give a complete
proof of the Poincare and the geometrization
conjectures According to the Fields medalist
Shing-Tung Yau this paper
was aimed at "putting the finishing touches to the complete proof of the Poincare Conjecture".

The true extent
of the contribution of Zhu and Cao, as well as the
ethics of Yau's involvement, remain a matter
of contention. Yau is both an editor-in-chief of the Asian
Journal of Mathematics as well as Cao's doctoral
advisor.
It has been suggested that Yau was intent on being
associated, directly or indirectly, with the proof of the conjecture and had
pressured the journal's editors to accept Zhu and Cao's
paper on unusually short notice.
MIT mathematician Daniel Stroock
has been quoted as saying, "I find it a little mean of [Yau] to seem to be trying to get a share of this as well."

In July 2006, John Morgan of

The above work seems to
demonstrate that Perelman's outline can indeed be expanded into a complete
proof of the geometrization conjecture:

Dennis Overbye
of the New York Times has
said that "there is a growing feeling, a cautious optimism that
[mathematicians] have finally achieved a landmark not just of mathematics, but
of human thought." Nigel Hitchin, professor of mathematics
at

## The Fields Medal and Millennium Prize

In May 2006, a committee of nine
mathematicians voted to award Perelman a Fields Medal for his work on the Poincare conjecture.
The Fields Medal is the highest award in mathematics; two to four medals are
awarded every four years.

Sir John Ball, president of the International
Mathematical Union, approached Perelman in St. Petersburg in June 2006 to persuade him to
accept the prize. After 10 hours of persuading over two days, he gave up. Two
weeks later, Perelman summed up the conversation as: "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." He went on to say that the prize
"was completely irrelevant for me. Everybody understood that if the proof
is correct then no other recognition is needed."

On August 22, 2006, Perelman was publicly offered the medal
at the International Congress of Mathematicians in Madrid, "for his contributions to geometry
and his revolutionary insights into the analytical and geometric structure of
the Ricci flow".
He did not attend the ceremony, and declined to accept the medal.

He had previously turned down a
prestigious prize from the European
Mathematical Society,
allegedly saying that he felt the prize committee was unqualified to assess his
work, even positively.

Perelman is also due to receive a
share of a Millennium Prize
(probably to be shared with Hamilton).
While he has not pursued formal publication in a peer-reviewed mathematics journal of his proof,
as the rules for this prize require, many mathematicians feel that the scrutiny
to which his eprints outlining his alleged proof have
been subjected to exceeds the "proof-checking" implicit in a normal
peer review. The Clay Mathematics
Institute has explicitly stated that the governing board which
awards the prizes may change the formal requirements, in which case Perelman
would become eligible to receive a share of the prize. [citation needed] Perelman
has stated that "I?m not going to decide whether to accept the prize until
it is offered."

## Withdrawal from mathematics

According to various sources, in
the spring of 2003, Perelman suffered a bitter personal blow when the faculty
of the Steklov Institute allegedly declined to
re-elect him as a member,
apparently in part out of continuing doubt over his claims regarding the geometrization conjecture. His friends are said to have
stated that he currently finds mathematics a painful topic to discuss; some
even say that he has abandoned mathematics entirely.
According to a recent interview, Perelman is currently jobless, living with his
mother in

He has stated that he is
disappointed with mathematics' ethical standards, in particular of Yau's effort to downplay his role in the proof and up-play
the work of Cao and Zhu. He has said that "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."
He has also said that "It is not people who break ethical standards who
are regarded as aliens. It is people like me who are isolated."

This, combined with the
possibility of being awarded a Fields medal, led him to quit professional
mathematics. He has said that "As long as I was not conspicuous, I had a
choice. Either to make some ugly thing" (a fuss about the mathematics
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.?