In November 2002, Perelman posted
to the ? the
first of a series of
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.
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
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:
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
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."