送交者: poem 2006年8月25日14:33:38 于 [教育与学术]http://www.bbsland.com

John Morgan: “In the mathematical context Perelman was forthcoming and patient”

John Morgan (Columbia
University, New York, USA,
together with Gang Tian
(Princeton University, USA),
has written a book presenting a
complete account of the proof
of the Poincaré Conjecture
based on Perelman’s ideas.
Morgan gave a press
conference yesterday at the
ICM2006. This is a
tranxxxxion of his answers.
About his lecture at the
ICM2006, entitled “Special
Lecture on the Poincaré
Conjecture”.
“Perelman’s work is an
extraordinary achievement. In
the last hundred years we have
learnt so much about topology
and as a result of that knowledge somebody with the
insight and technical power of Perelman could finish off
this fundamental problem. It’s a great success.
“Mathematics is a discipline in which there are very
hard problems that you don’t know how to solve, but if
you study more and more you can get slowly a deeper
and deeper understanding. This is what has happened
with the problem of the Poincare conjecture. To
paraphrase Newton, Perelman has seen far, but in order to
do that he has stood on the shoulders of giants. And
certainly one giant who stands out is Richard Hamilton,
who painstakingly over 25 years developed the basic of
this theory [the ricci flow] and laid the foundation for
Perelman’s work.
Are you satisfied with the work of Perelman, is there
a consensus in the community about the proof the
Poincaré Conjecture?
Gang Tian and I have written a paper of 473 pages on
what we consider a complete revision of the proof of the
Poincaré Conjecture: this is the result of several years in
which I tried to understand Perelman’s arguments. And as
every professor of mathematics knows, if you want to
learn a subject, teach it. At the end of that experience I
am really convinced that the Poincaré Conjecture is
proved. (...) There are also two other really long articles
by Zhu and Cao and another one by Kleiner and Lott
going over much the same terrain. And yes, I would say
that at that point you have a consensus among the people
who have seriously studied Perelmans’ preprint and Hamilton’s work that the
conjecture has been proven.
About the paper by Xi-Ping
Zhu and Huai-Dong Cao, and
the controversy about the
authorship of the solution of
the problem.
To me all three of these papers
play a similar role but not
exactly the same... but all of
them have to be viewed as
unpacking, expanding on, filling
in missing details. I am not an
expert on either of the other two
articles because I was so busy
writing my own. But I certainly
don’t see major significant
contributions independent of
what Perelman has done. And
after looking the paper by Zhu
and Cao it seems to me an honest account with the
appropriate credit all the way through, and all the
controversy stems not from this paper but from the noise
around it.
About the differences between the three publications
on Perelman’s work
Our work is more different from the other two than
they are from each other. The main difference is we
focus only on the Poincaré Conjecture, while those
articles discuss the more general geometrization
conjecture, and the arguments take different paths. The
main body of all three treat the same subject, which is
the existence of the Ricci Flow with surgery. If you want
to prove the Poincare Conjecture only, which is what we
chose to do, there is a third pre-print by Perelman, and
we followed that argument because we thought it was a
beautiful argument on its own, both Kleiner and Lott and
Cow and Yow do not treat that third Perelman paper, but
rather follow the suggestion at the end of the second
paper, which deals mostly with the geometrization
conjecture.
Is Perelman’s proof complete?
I believe if you take all three papers by Perelman
together they are 55 or 60 pages, and we wrote 473
pages. The first hundred pages were mostly background
inxxxxation and the rest was mostly unpacking and reordering
what was contained in the 55 pages of Perelman.
I think that Perelman was writing for experts in the field.
We are writing for graduate students.
The experience I have had multiple times when reading
Perelman is that I would read something and I wouldn’t
understand a word of it. Then I go home and think about
it. If I didn’t understand it I would talk to Tiang about it,
to Hamilton... When I eventually understood it –hours
later, days later, sometimes weeks later—I would ask
myself, OK, if I had to explain its main points as a guide
in one paragraph what would I do? So having had that
experience over and over again and never finding that the
paragraph that Perelman wrote deviated from an
absolutely accurate if incredibly compressed dexxxxion
of the argument that I had understood to be completely
correct, I conclude that Perelman had just decided for
some reason to compress everything.
In relation to the sentence in the abstract of the
paper by Zhu and Cao, where they talk about the
“Perelman-Hamilton” proof of the Poincaré
Conjecture.
“I guess that is their view of it”
Can we say that Perelman has proven the Poincaré
Conjecture?
Yes, I would say and I will say today that Perelman has
proven the Poincaré Conjecture. But one has to
understand that he would not have done it without
Hamilton’s work. Yet, in the culture of mathematics it is
my view that the credit for proving the Poincaré
Conjecture should go to Perelman.
What will change in the field after the
demonstration of the Poincare Conjecture?
In some sense nothing will change. In the work in 3D
topology the state of the art will not change (...) . The
significance is the understanding of these evolution
equations, the Ricci Flow equations and the way they
develop singularities. This will have applications in other
mathematical areas (...). And a little more speculative, the
way these evolution equations develop singularities is
incredibly important within mathematics and in physical
phenomena. Many, many physical phenomena are
explained by evolution equations similar to the Ricci
Flow, so understanding the way these equations develop
singularities, and how to treat these singularities, have
tremendous consequences inside mathematics and
physical sciences.
Have you met Perelman?
I met him when he made his tour to the United States in
2003, when he came to explain his ideas. I attended
several of his lectures and then talked privately with him
on several occasions, and after he went back to Russia I
was in e-mail contact with him when I was struggling to
understand his writings. And he was always very
forthcoming and patient in explaining his ideas, so while
he seems to be socially reclusive, in the mathematical
context he was forthcoming and patient.
Would you say he is a genius?
I don’t use words like that. He is an incredibly talented
and insightful and powerful mathematician. What he has
managed to do is a reflection of incredible power. I just
don’t find use in that kind of word.
How did you experience the controversy around the
authorship of the demonstration of the Conjecture?
I was mostly focused on trying to be in the position I
am today, sitting in front of you and saying that Perelman
proved the Poincare conjecture, so I was very focused on
having this 473 pages completely written and making
sure that I completely understood the arguments entirely.
I was of course aware of what happened, but I wanted to
stay focused.
What was the key point in Perelman’s argument?
What was it that Hamilton didn’t see?
Hamilton knew that you had to study singularity
development, and he proved several important properties
about the way singularities develop. But he missed one.
Perelman realised that this extra property, never
considered by Hamilton, was in fact essential. Hamilton
had one singularity; he was very worried about he called
the cigar. And what made everyone sit down and take
notices was that he established that the cigar couldn’t
happen. And that was a real problem in Hamilton. So
Perelman added a new condition on top of the others that
Hamilton had discovered, for the way singularities
develop. He recognised that this was an important
condition and by a completely different technique, unlike
anything in Hamilton, established this condition. And
then he went on to establish that he had very powerful
control of the way singularities develop, which enabled
him to do the kind of surgery that Hamilton had proposed
doing but couldn’t always carry out. That to me is the
most original insight.
But Perelman also had the incredible technical power to
take the tools that Hamilton designed and use them in an
even more subtle way to understand not only what one
singularity was but to control them all.
Will the Poincaré Conjecture now change its name
to Poincare’s Theorem.
I believe it will always be the Poincaré Conjecture. To
me the statement that Perelman has solved the Poincaré
Conjecture means Perelman has turned the Poincaré
Conjecture into a theorem. But we will still talk about the
Poincaré Conjecture, because the name is deeply involved
in mathematics.
But everyone will think that it is now a theorem...
I hope so! I hope everyone will read the 1000 pages
and say, of course!
Should Perelman get the millennium prize of the
Clay Foundation?
Unfortunately that is not for me to say.
You must have spent an incredible amount of time
on it.
I was motivated by three things. One is: I am a
topologist, and here is a solution to the most fundamental
question in the subject, how did this guy do it? I wanted
to understand for myself. And when I started to
understand the arguments I was more and more impressed
with the beauty of the argument. That was my second
motivation. Also, my third reason was to make a service
to the community, I didn’t want the topologists not to be
able to understand such a beautiful argument. Of course
had I known how much work was really involved, I
might have taken a different path two years ago!