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Perelman limit case: not a single paper, refuse the Field medal, refuse reviewers

Mike Ciavarella's picture

 

The crisis of journals, academic circles, and peer review is perhaps extremely clear explained by the curious and surprising case of recent history. 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.

See the attached

A
small contribution of the WSEAS Staff to GRIGORI PERELMAN

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 Leningrad (now St. Petersburg) on
June 13, 1966.
His early mathematical education occurred at the world-famous
Leningrad Secondary School
#239
, a specialized school with advanced
mathematics and
physics programs. In 1982, as a member of the USSR team competing in the International Mathematical Olympiad, an international competition for high school students, he won a gold
medal, achieving a perfect score. In the late 1980s, Perelman went on to earn a
Candidate of Science degree (the Russian
equivalent to the Ph.D.) at the Mathematics and Mechanics Faculty of the
Leningrad
State University
, one of the leading universities in the
former Soviet Union. His dissertation was entitled "
Saddle surfaces in Euclidean spaces"
(see citations below).

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.
Hamilton's basic idea is to formulate a "dynamical process" in which
a given three-manifold is geometrically distorted, such that this distortion
process is governed by a differential equation analogous to the
heat
equation. The heat equation describes the behavior of
scalar quantities such as
temperature; it
ensures that concentrations of elevated temperature will spread out until a
uniform temperature is achieved throughout an object. Similarly, the Ricci flow
describes the behavior of a
tensorial quantity, the Ricci curvature tensor.
Hamilton's hope was that under the Ricci flow, concentrations of large
curvature will spread out until a uniform curvature is achieved over the entire
three-manifold. If so, if one starts with any three-manifold and lets the Ricci
flow work its magic, eventually one should in
principle obtain a kind of "normal form". According to
William
Thurston, this normal form must take one of a small
number of possibilities, each having a different flavor of geometry, called
Thurston
model geometries.

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 Russia, he is said to have gradually stopped
responding to emails from his colleagues.

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 Columbia University and Gang
Tian
of the Massachusetts Institute of Technology posted a
paper on the ?titled, "Ricci Flow and the Poincare Conjecture." In this paper, they claim to
provide a "detailed proof of the Poincare
Conjecture". On
24 Aug 2006, Morgan delivered a lecture at the ICM in Madrid
on the Poincare conjecture.

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 Oxford University, has
said that "I think for many months or even years now people have been
saying they were convinced by the argument. I think it's a done deal."

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 St
Petersburg, and subsisting on her modest pension.

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.?

  • Professor Marcus du Sautoy of Oxford
    University
    has said that "He has sort of alienated
    himself from the maths community. He has become
    disillusioned with mathematics, which is quite sad. He's not interested in
    money. The big prize for him is proving his theorem.

A
small contribution of the WSEAS Staff to GRIGORI PERELMAN

 

 

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