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

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.

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 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.

**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.

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."

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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