Morgan at yesterday's press conference

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 transcription 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 of 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 Cao and Yao 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 information 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 Tian 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 description 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 realized 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 recognized 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!