Perfect Rigor

I just finished reading author Masha Gessen’s new book about Grigori Perelman, Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century. It’s a short but very well done account of the life of Grigori Perelman, how he came to prove the Poincare Conjecture, and what has transpired since.

The book is really not about mathematics, but about mathematicians and their culture, especially that of Russian mathematicians. Only one chapter deals with the mathematical content of the Poincare Conjecture, with the bulk of the book about Perelman and his career. Perelman’s talent’s were recognized early, and were nurtured in Leningrad by a system designed to train students for mathematical competitions. He won a gold medal at the International Mathematical Olympiad in 1982. The institutionalized anti-Semitism of the Soviet mathematics establishment of this period is described in detail in the book, together with the intense efforts made by Perelman’s supporters (including Alexandrov) to overcome this. He did his graduate work at the most prestigious institution in Leningrad, and then went on to a research position there at the Steklov Institute.

Gessen never managed to interview Perelman himself, but did talk to many if not most of the mathematicians he interacted with. He was brought to Courant by the intervention of Gromov, and for a few years worked there, at Stony Brook and at Berkeley. By the end of this time, he had started to develop a significant reputation in the math community, but he chose to return to Steklov and pretty much dropped out of sight, communicating with very few people for several years. It was during this period that he developed his proof, finally posting what could be described as a detailed outline in a series of three papers submitted to the arXiv.

The story of what happened then is rather remarkable, but it’s a story I’m pretty familiar with since I got to watch much of it from up close (Perelman’s preprints and the question of whether he really had a proof were discussed intensively here at Columbia, where Richard Hamilton and John Morgan are among my colleagues, and quite a few other people work in this area). Gessen does a good job of telling this story, adding some details I was unaware of.

Perelman turned down the Fields medal awarded him for this work, and sadly, he seems in recent years to have cut himself off from even his closest friends in the math community. Indications are that he is no longer actively working on research mathematics. The book contains speculation from several mathematicians who know Perelman about his thought processes and the reasons for his behavior, but they remain somewhat of a mystery. Some amount of paranoia seems to be at work, together with an intense distaste for any sort of politics, even the most innocuous workings of the mathematical community and its institutions.

The last chapter of the book has some news I hadn’t heard. Last year, Jim Carlson, who runs the Clay Mathematics Institute and is responsible for the process that will determine the award of the million-dollar Millennium prize for the proof of Poincare, traveled to St. Petersburg. He talked to Perelman on the phone, but Perelman refused to meet with him. According to the book, Clay was planning on convening a committee to decide on the prize this past May, with a report planned for August. Presumably this all has already happened by now, and perhaps Carlson has already made another trip to St. Petersburg in a last attempt to see if Perelman can be convinced to accept the prize. Perhaps we will be finding out the results soon…

Update: Today’s Wall Street Journal has an article by Gessen about Russian mathematics that summarizes part of her book.

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28 Responses to Perfect Rigor

  1. Giotis says:

    The poor, lonely, eccentric genius who proves the Poincare conjecture, denies the Fields medal, the money, the honours and the whole world and disappears in oblivion.

    This is the stuff that legends and myths are made of.

  2. tst says:

    “This is the stuff that legends and myths are made of.”

    And you think Perelman is unaware of that?

    The math he did is first rate, of course. But the drama adds nothing.

  3. milkshake says:

    tst: Its not a pose, he withdraws from things which he does not want to deal with. The unpleasant part of the drama was not his making anyway.

  4. Tom O'Bulls says:

    What finally happened re Yau and his legal case?

  5. ds says:

    Giotis: why do you think he is lonely? Some people can be quite happy on their own and even prefer it, and even choose it as a lifestyle.

    I find it infinitely irritating how the majority of people equate being alone with loneliness. Some people just aren’t so shallow that they need constant stimulation and distraction from themselves in the form of others and things. They may restrict their attention to just a few people.

    Perhaps Perelman sees a corrupt institution before him and chooses not to participate. Maybe this should make us stop and think;; instead we just marginalize him and say he is a “poor, lonely genuis,” which is saying don’t take him seriously, he is brilliant in math, but that same brilliance makes him completely incompetent in social matters.

  6. ds says:


    “And you think Perelman is unaware of that? ”

    Do you think he is really so petty as to be motivated by that? I don’t think so. You see, this is another attempt at marginalization. Society, the norm, sees an individual who refuses to play by it’s rules, and it tries to dismiss him, explain him away as mal-adjusted or petty. That way we don’t have to think, we can just carry on as usual; in our petty competitive way, chasing prizes, positions at prestigious universities, and so on.

  7. ds says:

    i should just add at the end of the last sentence of my previous message: “or whatever particular aspect of our behavior Perelman may be repulsed by..”

    Sorry for the SPAM!

  8. Giotis says:

    ds, I was referring to the way the general public might perceive his life and not to his actual reality.
    In any case I don’t give a negative tone to the word ‘lonely’. The idea of the lonely hero is omnipresent in literature and in popular culture, captivating human minds throughout the centuries.

    But yes, you are right; a loner is not necessarily lonely although an old saying alleges that if someone lives all alone and doesn’t feel loneliness, is either mad or god.

  9. Paul Titze says:

    Why not challenge Grigori with another mathematical challenge that’s suited to his skills with an even bigger prize? He might be so enticed by it that he’ll come out of the woodworks 🙂


  10. D R Lunsford says:

    Does any single person understand all the details of the P.C. proof?


  11. Sakura-chan says:

    The claymath website continues to refer to it as a conjecture. I wonder when they’ll dub it as a theorem.

  12. milkshake says:

    prizes are apparently not what he is after; maybe they should try to convince him to move to IAS

  13. Peter Woit says:


    Perelman appears to have no interest in a position at the IAS (or anywhere else for that matter).


    He seems rather offended by the whole concept of the prize he already has just about won, I don’t think another one would inspire him in any way.


    Perhaps the Clay web-site will be updated soon…


    The mathematicians who wrote up detailed versions of the Perelman argument (Morgan-Tian, Cao-Zhu, Kleiner-Lott) presumably understand the full details, and so perhaps do other people who have read these documents and worked in this area. The Morgan-Tian book was especially carefully refereed, by several of the best mathematicians in the world (ever wonder why Terry Tao has given talks on the proof and written about it in his blog?). I don’t know if these referees divided up the sections of the proof, or each of them went through the full thing.

    In any case, this proof now has been about as carefully examined as any.

  14. If I may make a guess, I think that Perelman would accept a position as a teacher for the very young, smart children, still uncorrupted from “politics” that invades everything and everywhere from the adult world. Although such a position could be seen by many people as an absurd downgrade (teaching kids is usually regarded as a minor profession by society), I think that he would probably enjoy such a fresh air. This is something that I think he needs. And such a “reboot” would perhaps motivate him to return working in mathematical problems again and have his results published independently in the arxiv.

    He has shown clearly not to be motivated by prizes or fabulous positions at all.

  15. Gigel says:

    Why is everyone trying to “fix” him? So he can become yet another useful tool for humanity, since if you don’t work for society, something must be clearly wrong with you? Maybe he plays world of warcraft all day and that’s what he enjoys. Or maybe he’s simply crazy. Why don’t we just forcefully hospitalize him and make him better, stuff him with meds until he agrees to hold a position at IAS and maybe prove 2-3 more theorems, right? That would make US feel better about ourselves, seing how well our society is working like an ant farm. Well what if he doesn’t give a crap about teaching or working in math anymore?

    Nobody owes anything to society, unless they personally feel they do.

    And btw, he never refused the million bucks. In fact, it was never offered to him. As far as he is concerned, he proved the conjecture, his work is done. If the institute is unable to solve its own policies, that’s not his problem. He just said he won’t discuss it or anything, because it’s done. Finito. Nothing to talk about. Whether it’s published where it has to be or not. Now give him the money or back out of the deal because of minor issues.

    He’ll never beg or ask for the money. That’s all.

  16. vincent says:

    I agree with “tst” that the drama aspect should have been irrelevant.

    Three other people got Fields medals along with him. He is getting the most publicity out of them all! Self-interest might not be the full story, but its silly to think that he has none of it.

  17. Tom O'Bulls says:

    “Three other people got Fields medals along with him. He is getting the most publicity out of them all! ”

    That’s because their work, outstanding as it is by normal standards, is utterly negligible compared to Perelman’s. That was really a bad year to win a Fields.

  18. Andy says:

    Tom O’Bulls — While I agree that Perelman’s proof of the Poincare conjecture was amazing, no one who knows anything about math would claim that the work of Tao, Okounkov, or Werner is “utter negligible” compared to it.

  19. milkshake says:

    Christine, you are right – a math Olympiad club, no pressure.

  20. Serifo says:

    I think Perelman and Grothendieck are examples of pure natural thinkers , any young physicist or mathematician should reflect about their stories .They are not motivated by international awards , fame , academic prestige or New York times magazine !

  21. Aristarchus says:

    “But yes, you are right; a loner is not necessarily lonely although an old saying alleges that if someone lives all alone and doesn’t feel loneliness, is either mad or god.”

    Why not both? Even the Bible says that God made humans for companionship and worship.

    Boy, he must have been lonely.

  22. abc says:

    I think that people should leave him alone. I guess he just loathes being a public figure. One cannot get the million dollar prize, Field medal etc. and than just disappear. Once you start to play the game, you are already deeply in it, so he refused everything from the very beginning. I would even go on and say, since it has become increasingly difficult for him to avoid publicity, he had to become more reclusive than he probably ever wanted

  23. observer says:

    Cantor, Godel, Sidis, Erdos,Grothendieck,Nash,Von Neumann, Perelman to name just a few of the great mathematicians and all of them had some very peculiar view of the world and some eccentricities to say the least, it seems to came with the profession itself.

  24. Deane Yang says:

    “They are not motivated by international awards , fame , academic prestige or New York times magazine !”

    And who is? Anyone who wants any of these things is not going to become a pure mathematician.

    I don’t really understand either the attacks on Perelman or the effort to put him on a pedestal. There is no question that he’s a great mathematician and that it is a loss for the rest of us that he has decided to turn his back on us. But he is not the only one who has done this, and his decision should be treated with respect. He’s just a human being trying to lead his life as best as he can.

  25. RMA says:

    Does Dr. Perelman even like people talking about his personal life (as opposed to his mathematical ideas) on public forums?

  26. milkshake says:

    … the paparazzis at his mother’s house and random freaks (snapping his pictures on subway with a cell phone) would be a more acute nuisance than some little Internet gossip

  27. Aristarchus says:

    I now consider him to be one of my few personal heroes. He did the math for the math and told society to go jump in a lake.

  28. Russ Van Rooy says:

    Sounds like shades of Alexander Grothendieck. A great biography of Grothendieck in English is long over due by the way.

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