About the title

About the title

I changed the title of the blog on March 20, 2013 (it used to have the title “Notes of an owl”). This was my immediate reaction to the news the T. Gowers was presenting to the public the works of P. Deligne on the occasion of the award of the Abel prize to Deligne in 2013 (by his own admission, T. Gowers is not qualified to do this).

The issue at hand is not just the lack of qualification; the real issue is that the award to P. Deligne is, unfortunately, the best compensation to the mathematical community for the 2012 award of Abel prize to Szemerédi. I predicted Deligne before the announcement on these grounds alone. I would prefer if the prize to P. Deligne would be awarded out of pure appreciation of his work.



I believe that mathematicians urgently need to stop the growth of Gowers's influence, and, first of all, his initiatives in mathematical publishing. I wrote extensively about the first one; now there is another: to take over the arXiv overlay electronic journals. The same arguments apply.



Now it looks like this title is very good, contrary to my initial opinion. And there is no way back.

Wednesday, August 13, 2014

And who actually got Fields medals?

Previous post: Who will get Fields medals in less than two hours?

Of course, if you are interested, you know already: Artur Avila, Manjul Bhargava, Martin Hairer, Maryam Mirzakhani.

I named in my previous post all except Martin Hairer, who is working in a too distant area in which too many people are working. I was put off tracks by the claim that M. Mirzkhani definitely will not get the medal. Before this rumor (less than a week ago) I would estimate her chances as about 60%. The award has no effect on my opinion about her work: her results are very good and interesting, but not "stunning", as it is said in the citation. Many people in related areas and even in the same area made comparable or much deeper and unexpected contributions.

I do not consider my estimates of somebody chances as predictions when the estimate is 60% or even 80%.

But I made three predictions, and they turned out the be correct: Artur Avila will be a winner; one of the winners will be a woman; one of the winners will be from Stanford. The first two of them were rather easy to made. But why Stanford? The idea materialized in my mind out of blue sky only few days ago; there was no new information, neither rumors, nor mathematical news.

Instead of a medal Jacob Lurie recently got a prize worth of 3 millions. I hope that he realizes that the decision of the Fields medal committee not to give him a medal tells much more about the committee than about the depth and importance of his work.


Next post: To appear

77 comments:

  1. According to Tim Gowers' blog, the committee for the Fields medals consisted of:
    Daubechies, Ambrosio, Eisenbud, Fukaya, Ghys, Dick Gross, Kirwan, Kollar, Kontsevich, Struwe, Zeitouni and Günter Ziegler.

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  2. There seems to be anti algebra bias judging by last two awards.

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  3. Anthony Leverrier: Many thanks. A little bit earlier than you posted your comment, I attempted to find this list by Google and failed.

    The list explains the choice of M. Mirzakhani over, say, Sophie Morel, or over Ursula Hamenstädt, working in the same area of mathematics (and not only, in contrast with Mirzakhani). Well, most likely Hamenstädt is over 40. This absurd age restriction (and the Fields did not included any age restriction into his will) hurts women much more than men. Apparently, at least some people started to realize this, but they not aware of the fact that it was finalized only in 1994 and can be reversed.

    Mirzakhani is the 3rd person who found a proof of that Witten's conjecture. It is hardly possible to know if this conjecture played a role in awarding Fields medal to Witten himself. This depends on when Witten announced it informally. A carefully written preprint (probably, the most accessible to a mathematician text by Witten) was widely circulating already about 3 months after the award.
    M. Kontsevich proved the conjecture before it was published (and that preprint was published very fast).

    The press release of the IMU about the work of M. Mirzakhani contains a remark about the Kontsevich proof which is to a big degree insulting:

    "Maxim Kontsevich (a 1998 Fields Medalist) proved Witten's conjecture through a direct veri cation in 1992."

    This is not even close to the truth. No "direct verification" of Witten's conjecture is possible. Kontsevich's proof is highly original, and was cited as the main reason for awarding him the medal.

    The next proof was by Okounkov-Pandharipande. A. Okounkov got a Fields medal in 2006. Probably, the next proof was by Mirakhani, 2014 medal (it is hard to assign precise dates to nearly simultaneous and independent works). Perhaps, the best proof is the one by Kazarian-Lando, who are not eligible because of the age.

    It looks like finding a proof of Witten's conjecture is the surest way to get a Fields medal (there is no need to be the first person to prove it, i.e. to be Maxim Kontsevich). :-)

    The president of the IMU (I. Daubechies) approached the task of giving one of the medal to a women very seriously (appointing this committee is the main responsibility of the president).

    It seems that giving one of the medals to M. Hairer was almost as important as giving one to a woman. Which is hard to explain, unless he has no serious achievements. My first impression was that he is an applied mathematician, but apparently he is not.

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  4. poslushnik: Since, apparently, you do not count Ngô Bảo Châu as an algebraist, you do not include the algebraic geometry and the number theory over finite fields into algebra. If so, there are only 2 or 3 persons who got a Fields medal for work in algebra: J. Thompson (1970), E. Zelmanov (1994), and, perhaps, R. Borcherds (1998) - anyhow, he proved a conjecture of J. Thompson.

    The bias against algebra is a very old phenomenon: the fates of Abel and Galois may serve as witnesses. Or the widespread dislike of S. Lang's "Algebra".

    M. Bhargava is the first medalist working in algebraic (in contrast with analytic) number theory, notwithstanding the spectacular development of the algebraic number theory for many many years (over 200?). Another bias... I was told that algebraic number theorists always avoided pushing their young colleagues for the medals, realizing the inherent danger for the medalist and for the area. I have no idea to what extent this is correct. I mean, not pushing. The danger is obvious.

    The most regrettable bias is the bias against "theory-builders" as opposed to "problem-solvers". It seems that only two times the medal was awarded for a new theory: to L. Schwartz (1950) for the theory of distributions, and to A. Grothendieck (1966) for the "new" algebraic geometry. Even when the medal was awarded to J.-P. Serre for mostly theoretical work, the committee was especially impressed, according to H. Weyl, by specific applications to the computation of homotopy groups of spheres.

    Of course, the age restriction, even before its formalization, pushed and continues to push toward problem-solvers: only very rarely it is clear from the very beginning what the fruits of a new theory will be. After we see the fruits, the mathematician who build the theory is not eligible because of age. For few decades there was enough of theory-builders who found specific applications of their theories very early. Now the medalists are just very good at using established theories. Sometimes they just proving new theorem faster than others, and hence have more theorems proved.

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    1. As far as I understood, the Fields medal has been primordially established as an award for 'problem-solvers' (just because there was actually no 'theory-builders' by 1930), and we should consider as a much more grievous bias the objection against 'theory-builders' by jury of Abel Prize. Am I wrong?

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    2. How did you arrived at such a theory? The standard description is as follows:

      "The Field Medals were first proposed at the 1924 International Congress of Mathematicians in Toronto, where a resolution was adopted stating that at each subsequent conference, two gold medals should be awarded to recognize outstanding mathematical achievement. Professor J. C. Fields, a Canadian mathematician who was secretary of the 1924 Congress, later donated funds establishing the medals which were named in his honor."

      There was more than enough "theory-builders" in 1930. I do not quite understand what do you mean "by 1930". But anyhow, there was more than enough in 1930 sharp. And one of the first two winners (1936), Lars Ahlfors, was a theory-builder.

      Abel prize remains much better than Fields medal in all respects. In particular, there is an algebraic number theorist among the winners. And he is a "theory builder", as are some other laureates, bypassed by the past Fields committees.

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    3. By the words 'outstanding mathematical achievement' most people I know imply 'solution of a complicated or long-standing problem'. The 'theory-building' itself worth nothing in their opinion and can be rewarded only if it leads to the solution of some problem. I thought that such view was completely conventional during the pre-Bourbaki era, so it would be rather naïve to expect that Fields medal will be given only to theory-builders.

      (I apologize for my typos and poor English)

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    4. deevrod: Nobody suggests to reward only the theory-builders. And nobody is interested in theories which are just theories, unrelated to existing problems. But problems solved by well established tools are not particularly interesting. Actually, a problem is really good only if attempts to solve it lead to new theories. And, as M. Gromov said, we don't know if a problem is good or bad until it is solved.

      In addition, there are different kinds of problems. Some problems are specific questions with yes/no answers. Poincaré considered such problems as uninteresting, definitely before 1930 and Bourbaki. His position was shared by V. Arnold, who never expressed any sympathy to Bourbaki. Other problems are "open" and asking, for example, for understanding of some phenomenon. You cannot "solve" such a problem without building a new theory.

      A remarkable example is the Monstrous Moonshine conjecture(s), relating the Monster simple finite group with the theory of modular forms. Here there a both problems: to prove the conjecture, and to understand why they are true. The conjectures are proved, and for the last step R. Borcherds got a 1998 Fields medal. But his proof was not recognized as an explanation why the conjectures are true. This problem is open.

      Back in 19 century one of the main problems was the problem of building foundations of mathematical analysis. If building these foundations is not an "outstanding mathematical achievement", then I don't know what is.

      There is a drift now toward very specific problems. I see a sign of crisis in this. For the time being, the potential of the theories built in 1935-1975 is far from being exhausted. But it will not last forever. Of course, nobody can be sure that the mathematics will last forever. I discussed this in some the first posts in this blog.

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    5. I do consider Bảo Châu and Bhargava to be algebraists, as well as Okounkov. I just think 1/4 of awards is not an accurate representation of significance of algebra in mathematics. Especially when there are worthy candidates.

      One theory builder (and problem solver!) who should have got it in the past is G. Lusztig. Now for a younger mathematician in his area it may be hard to get it, since its very hard to be even better than Lusztig.

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    6. poslushnik: Well, we may discuss the question "what is algebra?" on some occasion. I got an impression that you are using a very narrow definition. You do not, but this still does not mean that pure algebra should be ignored. And I would not agree to count Okounkov as algebraist anyhow.

      G. Lusztig is a good example. Apparently, a lot of people were more or less sure that he will get 1990 medal. In fact, not a single person working in representation theory got a medal. But this is, perhaps, half of the mathematics (pure) - if we take a sufficiently broad definition. Harish-Chandra, Langlands - no medals. And the Langlands program is one of the main branches of mathematics now.

      I am not inclined to agree with the "good representation" argument. It is extremely dangerous. Be assured that 80% of papers in mathematics are not devoted to algebra. They are devoted to differential equations. The people who wrote these papers do not know that algebra exists. If such a mathematician knows that the LFT was proved, it is already good.

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    7. According to wikipedia, Lusztig was born in 1946, so I do not see how people could've been sure that he would get the 1990 medal. He did recently win the significant Shaw Prize; and I think he is an excellent candidate to be awarded the Abel Prize at some point.

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    8. shopkins: Sorry. This means that either I, or these people were mistaken. May be it was in 1986 (and only 3 medals were awarded, as far as I remember), may be these people didn't knew when he was born. There was no Wikipedia, no internet - it was hard to know the age of anybody. And the rule wasn't fully formalized till 1994.

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  5. "I was told that algebraic number theorists always avoided pushing their young colleagues for the medals, realizing the inherent danger for the medalist and for the area."

    I was wondering why is the medal considered a threat for the area?

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  6. Bourbakifan: I don't think that it is always considered as a threat. Obviously, people usually pushing younger people in their area for all possible awards. I quoted a personal opinion of a number theorist - a well informed number theorist. I don't know to what extent he was right. He may be wrong, but it is the fact that Bhargava got the first medal for algebraic number theory. At the same time, there was no shortage of extra-class mathematicians in algebraic number theory. Very often some of them were members of the Fields medals committee, but this never (up to now) lead to an award.

    I can offer a lot of arguments in support of the position of that number theorist. Fields medals corrupt the community. They distort mathematics - the choice is often random, 2-4 out of 40 of about the same level candidates.

    But, perhaps, an example will be more convincing. Take the topology of smooth manifolds. Not very special theories in the dimension 3 and 4, but the general theory. Fields medals in 1958, 1962, 1966, 1970. And that's it. Soon after this the field is dead. All Fields medalist left the field. The best people younger also left it. Many started to talk that all problems about smooth manifold are solved. But that's far from being true. Some started to say that there is nothing interesting in this theory. I don't believe in this either.

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    1. I find it hard to believe that awarding a Fields Medal is detrimental to an area. Perhaps there are other reasons for topology of smooth manifolds loosing popularity?

      It seems to me that in the modern age having the Medal gives an extra weight to the letters of recommendation written by the person, and thus to his/her whole area.

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    2. poslushnik: One can find a thousands reasons. Starting with claiming that topology of manifolds never was interesting. Of course, the medals are not the main reason for topology of manifolds going out of fashion.

      But, apparently, it was very important in the defunct by now superpower. S. Novikov is a great mathematician and a great person, but he was a topologist, got Fields medal for his work in topology and then took all his best students to the mathematical physics. He repeatedly said and wrote that one should not waste your life on topology or, whatever, some algebra. One should work on real things: theoretical physics. OK, one country out of development of topology of manifolds. He is an authority, who knows better than him, if this topology worth pursuing?

      You are using the only argument which was suggested in support of introducing the institute of Fellows of the American Mathematical Society. I believe that having Fellows is very detrimental.

      Fields medals are much more detrimental. There are 4 persons who are selected by a secret political process - we can assume that it is random - out of 40 mathematicians of approximately the same level, working in areas of more or less the same importance. Why these 4, or the people for whom they will write letters, should be given an advantage over other 36? Imagine, you did something remarkable about finite groups. The last and the only time when somebody working in the theory of finite groups got a medal was in 1970. Who will write you the letters?

      In fact, somebody will, but you will constantly see other people bypassing you. Like you see in the "Gowers's section" of the congress.

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    3. I agree about the Fellows program.

      In case of Fields medal, I also agree with what you are saying, but I would feel sad if it was not there at all. There is something romantic if you wish about acknowledging contribution of really great mathematicians.

      It would be nice if Fields medal was reformed. As you say, a transparent selection process would be nice. Even announcing a shortlist could make a difference. If we knew the committee's short list of 8-10 people, or whatever length it is, we would easily see all the political biases involved in the process. Its one thing to not give medal to Lurie pretending he just does not exist, its another to show that you considered both him and Hairer and chose Hairer.

      Another reform could be increasing number of medalists.

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    4. poslushnik: Grisha Perelman said: the solution of that problem is a reward by itself. Any prize is negligible compared to this. Why not to acknowledge great mathematicians by reading their papers, developing their ideas, just admiring them?

      I understand you, I myself used to have some romantic feelings about it. But things changed. Fields medal is beyond repair. We have to continue to award it, because of the Fields's will. The best thing to do about it is to stop paying attention. But this seems to be impossible. Let's drop the "no more than 40 in the year of congress" restriction; this should be possible.

      As of rewarding great guy, think about such issue: no matter how good you arrange things, always there will be people on the border, who did not get the award by an accident. The will be even people much better than most of the winners, who did not get the prize. Such people often feel insulted and are quite bitter. This is no good. For me, this alone is enough to get rid of all prizes.

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  7. "It seems that giving one of the medals to M. Hairer was almost as important as giving one to a woman. Which is hard to explain, unless he has no serious achievements. My first impression was that he is an applied mathematician, but apparently he is not."

    Sowa, what did you mean by this?

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    1. Jamie Vitara: I don't quite understand you. I wrote what I wanted to say. There are several statements in these three lines. All of them are tentative due mostly to the secrecy practiced by IMU. Which one is unclear? I doubt that I will be able to answer your question, unless you explain it.

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    2. Sowa, it is not clear at all what you were trying to say. Why was giving a medal to M. Hairer almost as important as giving one to a woman? In what sense? Why is it hard to explain, unless he has no serious achievements? Did you mean that , if he had no serious achievements then it would be easy to explain but, because he does have serious achievements, they didn't really need to try so hard to give it to him, because he deserved it anyway? The entire statement sounded very cryptic to me, maybe, because you didn't really say what you wanted to say.

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    3. Jamie Vitara: Thanks. Now I understand your difficulties better. But you did understand me correctly. Yes, “if he had no serious achievements then it would be easy to explain but, because he does have serious achievements, they didn't really need to try so hard to give it to him, because he deserved it anyway?”

      Still, I will try to explain some details and the background, but I have serious doubts that I will be able to do this within a comment, and even that this would make sense. The point is that this blog is not a collection of independent posts. There are several interwoven treads, and several of the first posts set up a background for most of them. I do not think that it would be reasonably to repeat in some condensed form the ideas discussed in details before. Fortunately, this particular tread is short - it was started about a year ago.

      My reply is very long, but I do not want to make a special post out of it. So, it will be split into two or three comments (Google does not allow too long comments).

      I have to start with a distant past. In 1954 Fields medals were awarded to J.-P. Serre and K. Kodaira. If you look at the composition of the committee, this choice is anything but expected. If you were well versed at the time into what is going on inside of the Institute for Advanced Studies in Princeton, you perhaps knew that the chairman of the committee and the most respect mathematician at the time, H. Weyl, was quite interested in the works of Kodaira and guess that Kodaira has a chance. But he was not the only member of the committee, and I am not aware of any hints that he could be interested in new French algebraic topology. Only one committee member, H. Cartan, definitely knew about Serre's work. The point is that representatives of the "classical" mathematics unanimously voted for awarding the medal to two mathematicians who just started to develop completely new methods. These methods are still not universally accepted, and are disliked by many.

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    4. The situation changed dramatically over the past 15-25 years. Now it is not hard to see the correspondence between the members and the winners. There are, say, 8 members and 4 medalists, and for each of medalists there are 2 or 3 members for support: they are either from the same area of mathematics (using much more fine division that Algebra-Geometry-Analysis), or they are from the same country, or something else.

      I claimed in this blog both a year ago and this year that I would be able to guess the winners if I knew the composition of the committee. Well, may be I was overconfident or assumed that some traditions would still be in place. The main relevant tradition is to award Fields medals only to pure mathematicians (well, E. Witten is ever not a mathematician, but he made some contributions to mathematics, and this was very pure mathematics). I definitely would be able to guess an applied mathematician, at least not by a glance on the list of members. I would not be able to guess M. Hairer at the spot. Perhaps, I would be able after doing some research with the help of Google and MathSciNet. The chances of success depends on how applied he is, and if applied, would it be the Computer Science (much more chances) or traditional ODE-PDE style applied mathematics (much less). I am not inclined to explain now the reasons beyond saying that I extremely rarely find an applied work to be of comparable with the pure mathematics depth.

      There are 4-5 members of the committee for supporting Hairer. I am sure that the main concern of the President of the IMU was awarding a medal to a woman, but I doubt that she has enough background to understand what the work Mirzakhani is about, or what are the works of Avila and Bhargava about. The same about 3 other members of the committee. One more is very far from the works of any of the winners, but he may (only may, I am far from being sure) have others reasons to support Hairer. There is no such a definite support for Mirzakhani. After looking at the list of the committee members, it seems that a specific support wasn’t needed. Everybody knew and accepted that one of the winners will be a woman, no matter how good she is, either absolutely or relatively to the other winners or other workers in her area. And only few cared about who will be this woman.

      Anyhow, knowing that awarding a medal to a woman was a priority (nobody I knew doubted this since 2010); the excessive support for Hairer seems suspicious. Another priority of the establishment is to award medals to applied mathematicians or at least looking like applied ones. This is to a big extent due to the desire of the various governments funding agencies to shift all mathematics in the applied direction (this is also a recent phenomenon – in 1960 the US Office of Naval Research funded works in differential topology). If an applied mathematician is really good (well, like Claude Shannon was), there will be no difficulty in awarding her or him a medal and no need for special support. This immediately led to two questions: is M. Hairer applied or pure mathematician? How good is his work? I don’t know the answers. If they emerge later, I will take a note. If not, my curiosity would be limited by the downloading a couple of his papers the day before yesterday. They turned out to be more interesting than I initially suspected, but far from being interesting enough to stimulate me to go further. I will better study the work of Mirzakhani. Or, even better, of J. Lurie – but the papers of Mirzakhani are much more accessible simply because they are much shorter.

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    5. In addition to the depth issue, I find giving medal to an applied mathematician disturbing for the rich kid / poor kid reason. Applied mathematicians have bigger salaries, a lot more grant opportunities, as well consulting with industry opportunities. Hairer for example writes and sells software.

      Giving Fields Medal to an applied mathematician is like taking arts endowment and awarding to an industry person.

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    6. poslushnik: The monetary value of Fields medal is negligible: about $15,000.00. Fields medal to Hairer will not change neither his, nor Lurie's access to grants and positions. They both already have what they need.

      The bad thing is not on the level of an individual. They are redefining the meaning of word "mathematics". There was a tradition of awarding Fields medals to pure mathematicians. Even Witten, who is not a mathematician, is rather a pure mathematician than applied.

      The main issue it the title of an article by late P. Halmos: "Applied mathematics is bad mathematics." He was a smart guy: on the first page he writes that he does not think so and simply picked up a catchy title. In the rest of the article he explain why the statement in the title is true.

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    7. Saying that Hairer is an applied mathematician is like saying that Ito or Kolmogorov or Varadhan are applied mathematicians. Hairer seems to be just as good (okay, maybe, not quite at the level of Kolmogorov, but he is still young). So all the statements about applied mathematics are irrelevant here. The fact that Hairer created successful music editing software is simply a testament to how versatile he is. The guy is brilliant.

      On the other hand, I agree that somebody like Lurie deserve the Fields medal just as much, if not more. So it would be best if there were no medals at all, or no 40 year rule, or give medals to ten people instead of four. The way it works now is terrible very unfair.

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    8. I agree, redefining mathematics to include applied is the worst thing that is happening. Its very sad that this trend got to Fields Medals. I still stand by my analogies rich kid/ poor kid, or art / industry. They just apply to areas as a whole, not specific individuals.

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    9. Jamie Vitara: I still don't understand your point. I said that I don't know if Hairer is an applied mathematician or pure.I am not going to rely just on your words - you did not supported them in any way. But, for the sake of the argument, let me agree that he is a pure mathematician. Then my conclusion is that he is very weak by the current Fields medals standards, but it was necessary to give him medal by some political reasons. I explained the reasoning above.

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    10. poslushnik: OK, let's use analogy art/industry. This would be not even an analogy. This how the things are. Mathematics (pure mathematics) is an art. Applied mathematics is not really mathematics. As somebody said and Arnold quoted (or it was Arnold himself?), there is no applied mathematics, there are only applications of mathematics. So, these applied mathematicians aren't working in mathematics. They know some mathematics, and work in industry.

      You cannot transfer resources from art to industry or vice versa. Moreover, it is much better for an art to be independent from anything external to it. Initially the government started to give away some money to some abstract fields. Some people, for example the late G. Mackey, predicted that eventually government will dictate what to do. It does for 25 years at least. The rising of wavelets started with NSF nudging the analysts to move from harmonic analysis to NSF. What we see now? A representative of this field was the president of the IMU during the last 4 year, and determined who will get these medals. I attended once her talk. It was long ago, and I had no idea about all this politics. But the talk was so boring... I avoided her talks since then, but tried once to listen to her husband, an "applied mathematician" (actually, he was working in industry, not in academia). It was also boring.

      If we will try to get "our" portion of the government and/or industry money, they will buy us, and tell us what to do. And it is well known what they want us to do: to beg for money. I was shocked when I learned that a leading computer scientist at Harvard has to spent 80% of his time writing grant proposals.

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    11. Sowa: Of course, you don't need to rely just on my words. But then I am very curious - how do you know he is weak, if you don't know anything about his work? You can not even judge if he is pure or applied, right? Then you can not judge how interesting the questions are that he is studying. Notice, I didn't say problems, because what he did was to develop a theory rather than just solve problems. Did you ask somebody at the level of, let's say, Varadhan? Or you do not consider somebody like Varadhan a great mathematician? Actually, I wonder if you consider Kolmogorov a great mathematician?

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    12. Jamie Vitara: I said already more than one time that would he be a good pure mathematician, then there would be no need to include 4-5 persons in the committee to support him.

      You see, already the fact that I don't know anything about his work suggest that he is applied. Actually, I tried to keep discussion by the formal circumstances like the composition of the committee. But I did looked at his papers. They do not *look* like a good pure mathematics.

      Why should I ask somebody of "level of Varadhan"? Why suggesting people in probability? You see, when I was very young, a lot of people from all branches of mathematics told me about Kolmogorov. May be I prefer to ask somebody of level of Serre? I know whom to ask. A really great mathematician once told me that there was nothing interesting in analysis between Riemann and Schramm. He is a very opinionated guy, but I managed to convince him that there was at least one really interesting theorem in between. I believe that I even don't need to ask.

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    14. I hope that if you asked Serre he would be humble enough to say - sorry, Sowa, I am not an expert on Hairer's work to judge how good it is, but I trust the opinion of great experts like Varadhan. If the Fields medal committee consists of 12 people, why is it unusual to have one probabilist, and two analysis/PDE experts? Did Avila win only because Ghys was on the committee? By your logic, there are definitely more people on this committee who should have supported Jacob Lurie over Martin Hairer (a.k.a. pure mathematicians)! As I said above, I would love to see Jacob Lurie get the Fields medal, but your logic is faulty, Sowa.

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    15. Jamie Vitara: Please, tell me what you are trying to achieve in this tread. I am at loss. Looks like you are trying to convince me that Hairer is (a) pure mathematician, and (b) an exceptionally good pure mathematician. If this is the case, you selected a wrong strategy. You can do this in only one way: by explaining me what he did. You don't need to tell me that writing some soft is not a problem. You need to tell me positive things. You may give references, accessible to a pure mathematician with a not big, but still more than average background in probability theory (note that in the USA universities probability is not a branch of mathematics from the administrative point of view; usually probability has its own department, independent of the department of mathematics). If there is an opinion of Varadhan, then I would like to read it. You are asking me not to trust Varadhan, you are asking me to trust your words about the opinion of Varadhan. I am not accusing you of a misrepresentation of his opinion. Frankly, I don't care about the opinions of either Varadhan or Serre about Hairer. What I care about is how either of them will motivate and support his opinion. For example, you are almost correct about Serre: publicly he will say something like you suggested. He is indeed humble enough and more than polite. What he may say privately is a completely different matter.

      My intention with respect M. Hairer was to defer the judgment till some sufficiently detailed description of his work will appear. (If there are video at the IMU site, it will not help me: only once I managed to watch more than 10 minutes of a recording of a mathematical talk.) You are forcing me to form my opinion now. Fortunately or not, but I never accept a point of view under pressure of any kind. The more you will argue in your current style, the more negative my opinion will be.

      Yes, it was essential to have somebody like E. Ghys on the committee to award a medal to Avila. In his case one person could be enough, everybody loves Avila. But, still, there was two persons to support him.

      Analysis and PDE are quite different branches of mathematics, especially if we are talking about PDE's of this sort. And yes, it is highly unusual to have more than one person from this area on the committee. Normally, it is sufficient to have just 2 persons sympathetic to the area. This time we have 4, plus one more who is there definitely not to promote any of the other three winners.

      By my logic (this is not mathematical logic, so, please, do not tell me what conclusions I should arrive at), 3 medals for pure mathematician is enough for other people. You would love to see Lurie to get the Fields medal, but he will not: his medal was given to Hairer. There are only 4 medals. 1 medal was reserved for a woman; I suspect the choosing one was the main topic of all deliberations. The choice is very good despite all my comparisons of her work with the works of other people in related areas, including women. 2 other pure mathematicians are the least controversial choice possible. This leaves only 1 slot potentially open for competition. If, say, Ib Madsen would be member of the committee (an exceptionally good candidate to represent both Europe and, say, any sort of topology, except dimensions 3 and 4, and any sort of abstract ideas), and the Hairer support group would be reduced to the usual 2, then the medal would went to Lurie. Note that Ib Madsen is not responsible in any way for this claim; it is just my guess based on his work only.

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    16. Jamie Vitara: Let me quote a technical (i.e., not expository paper for general public) paper by M. Hairer:

      "In Section 2, we provide a more detailed mathematical formulation of the main results of this article and we explain the main ideas arising in the proof."

      A (pure) mathematician would never write in a research paper that some statement is "mathematical". She or he may point out something opposite, like "for the benefit of people using ..., we include a description of the main results accessible for experts in ... ".

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    17. Sowa: Look, I agree with you on almost everything. Yes, Fields medal is bad for mathematics. Yes, 40 year rule is bad. Yes, they should have given it to Jacob Lurie. My only objection is that you are saying that Hairer is not a great mathematician. All of the above is not his fault.

      What he did was to give a definition. Before him, nobody knew what it means to be a solution of various stochastic PDEs arising in physics. His definition was unexpected and probably inevitable. And like Quillen, he did not solve any problem in the traditional sense. You said that Quillen's ideas appear to be "summoned from the void". If you read the Quanta Magazine article, you will see that mathematicians in Hairer's field had the same feeling about his work. So all I am trying to do here is to say that, just because we are unhappy about the way the Fields medal is awarded, we should not release this frustration onto somebody convenient.

      As for your quote from Hairer's paper, to me it is equivalent to the following quote from the paper of Kontsevich and Manin on "Gromov-Witten classes, quantum cohomology, and enumerative geometry": "Together, §2 and §4 can be considered as a pedagogical attempt to present the formalism of correlation functions of topological sigma–models in a form acceptable for mathematicians with algebro–geometric background." It is essentially the same, since he was talking about a problem coming from physics.

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    18. Jamie Vitara: The word "great" is quite serios. Riemann was a great mathematician, Poincare was, Hilbert was. I did not even attempted to counter such claims. My claim was: either he is not a (pure) mathematician, or he is not of the Fields medal level. Thanks to your efforts, now I am inclined to think both of these statements are true. This is not final, of course. I see no need to hurry.

      Sorry, I cannot believe that "nobody knew what it means to be a solution of various stochastic PDEs arising in physics." What all these people were doing? It is a huge field, stochastic PDE.

      Quillen did solved a couple of problems (of yes/no type) which were considered to be central problems at the time. And some other not so prominent.

      Cannot agree. Kontsevich and Manin speak about making their ideas accessible to mathematicians with a particular background. They do not call any statement in their paper "mathematical" - the paper is in a mathematical journal. If a paper would be published in a physics journal, then it would be reasonable to say that this is something "mathematical" (i.e. the reader may ignore it, but the authors were not able to resist), and something is not.

      Hairer uses the word "mathematical" in the situation where it really not admissible, and gives himself away. He is not a mathematician, pure or applied (I am not sure that the notion of "applied mathematician" makes any sense).

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    19. Sowa: Because stochastic PDE is such a huge field, it makes Hairer's work even more impressive. Indeed, for many non-linear SPDE there was no theory of solutions. I am sure the phrase "What all these people were doing? It is a huge field!" applies to your field of mathematics as well.

      http://gowers.wordpress.com/2014/08/18/icm2014-hairer-laudatio/

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    20. Jamie Vitara: No, it does not. The most impressive achievement is the creation of a new field. Then there are only few people working in it by the time of the award. Examples of such Fields medalist in the past: J. Milnor (1962), M. Atiyah, P. Cohen, A. Grothendieck, S. Smale (1966),G. Margulis, D. Quillen (1978), W. Thurston (1972). The list is not meant to be complete.

      That phrase does not applies to my field, and I am sure it does not applies to the stochastic PDE's either. People are studying such PDE's for more than half a century, in particular, their are studying their solutions, and they don't know what a solution is? I am not inclined to look for an introductory graduate textbook and giving you a reference. It is you who made a highly unusual claim (people studying equations don't know what a solution means), and therefore the burden of proof is on you.

      In my field, or, rather, fields of mathematics, people do know what they deal with.

      I am surprised that you give a reference to Gowers in a blog with a title like my. But, after I managed to find the relevant part of the post, which is strangely placed at the end, I found his outline useful. Even Gowers admits that all this looks like trivial ideas. So it looks for me too. The modifications made by Hairer are also quite natural, as is the fact that he needed to modify some standard techniques. Everybody doing something moderately interesting does this.

      I am not claiming that Hairer had no new ideas of some interest. It seems that he is a good semi-mathematician (like some people working in string theory are sometimes called semi-physists) who mastered a fairly difficult branch of mathematics (but many people working for the Wall Street mastered this branch too), and is fairly productive. It was fairly amusing to look at his citation data on MathSciNet. They are very low for a mathematician working in such an expansive field. Any good analyst has better numbers. Please, do not think that I am using these metrics uncritically, as a typical dean will do (I immediately made adjustment for the field). These data are useful only if you are using them in a context and with a lot of reservations. I always keep this in mind.

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    21. There is a typo in the above post. Thurston was awarded Fields medal in 1982.

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    22. Alex Reed: Thanks, of course, in 1982. I cannot fix it there: the comments cannot be edited.

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    23. Dear Sowa, Could you please clarify "A (pure) mathematician would never write..." statement above? Admittedly, Hairer works on the subject which has been (previously) much more discussed in the physics literature than in the mathematical one. Does this fact alone make him a "non-mathematician" or "applied mathematician"? Otherwise it seems natural to stress the level of rigor in the work, especially if it is one of its main points.

      Arnold included the word "mathematical" in the title of his book on mechanics, in a rather similar situation: mathematical writing on a subject traditionally treated by physicists.

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    24. mtsyr: A (pure) mathematician is working in mathematics, and everything he does as a mathematician is "mathematical" by default.There is no need to add this qualification. Quite the opposite, if a research paper contains something not belonging to mathematics (for example, motivation, or not rigorous by the current standards arguments for the benefit of general public), then this should be and will be pointed out.

      There are no levels of rigor in modern mathematics, and nobody stresses rigor. If the rigor is the main point, the paper does not belong to mathematics. Frankly, I doubt that this is the case with that paper. It is a typical technical paper devoted to some advances in a relatively sophisticated branch of analysis.

      The title of Arnold's book is "Mathematical methods of classical mechanics". Classical mechanics is not a branch of mathematics, and not all mathematical methods are useful for the classical mechanics. So, the title makes a perfect sense and does not disqualifies him as a mathematician.

      The qualification "mathematical" would make sense in a physical, biological, etc. journal. It does not make sense in a journal publishing only research papers in mathematics (this quote is from such a journal).

      Of course, Hairer wrote in the style he used to, and his style is to point out something mathematical. Hence he is not a mathematician.

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    25. Could you point out the difference between Arnold's case and this one? A major part of the theory of KPZ equation is even less a branch of mathematics than Classical Mechanics. Not all mathematical methods are useful for KPZ equation. So your reasoning "exonerating" Arnold (or e. g. Sinai with his "Theory of phase transitions: rigorous results") perfectly applies to Hairer.

      Your remark about the style indicates that while this paper is mathematical, many of other papers of Hairer are not, which shaped his style of writing. Is this a correct understanding of your claim? Anyway, you did not explain why personality and habits of Hairer are the only possible explanation of the appearence of the word "mathematical", and why existence of waste non-mathematical literature one the same subject, or if you wish, about the same object, could not serve as such an explanation.

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    26. mtsyr: It seems that you know Hairer's work well. Did you read Arnold's book? It is much more elementary and accessible. If you did, the difference should be obvious to you. Anyhow, the most obvious difference is the fact that the Arnold's book is a book, and Hairer's is paper is a research paper in a journal devoted to research paper's in mathematics only.

      I wonder, why do you care about this, apparently, minor issue. Why it is important for you to classify Hairer as a mathematician? (Why it is important for me not to classify him as a mathematician is already explained in the comments here.) I wonder, why a chemist or an expert in middle school teaching may want to be called a mathematician? This is a complete mystery for me, but I know this for fact.

      Fields medal was given once to a physicist. This did not made him a mathematician, and he, to the best of my knowledge, never wanted to be classified as a mathematician. His contributions to physics are highly controversial, but they are real, as is his influence in mathematics.

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    27. My question was very simple: why the reasoning you applied previously to Arnold does not apply to Hairer as well? My undestanding is that you withdraw that reasoning and offer a new one, that such a clarification is nesessary for a book (albeit written by a famous pure mathematician), but unsuitable for a pure math journal. OK, let's discuss this point. First, there are tons of counterexamples:

      http://link.springer.com/article/10.1007/s00222-005-0444-1
      http://link.springer.com/article/10.1007/s00222-006-0022-1
      http://annals.math.princeton.edu/2013/177-2/p07
      http://annals.math.princeton.edu/2012/175-3/p14

      Second, years before a paper is published in Annals (which puts all nesessary flags in place), it circulates as preprint.
      Third, there are excellent journals (e. g. CMP) that accept both physics and math papers, and at the time of writing the author may not know which journal he will choose. The final point is that even if the clarification might appear redundant, it is hard to see how can it hurt.

      I don't think my motivation to answer your post is of any interest. The "minor issue", along with the composition of the prize commetee, was apparently the sole basis for some quite strong statements you've made here.

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    28. mtsyr: I am sorry, but I am too lazy for transforming your links into links which will allow me to access these papers from this computer.

      The rest is irrelevant. Preprint is a preprint, and before submitting a paper it is adjusted to the journal. This can be done after submitting too. Minor changes, like removing the word "mathematical", can be done even after the paper is already typeset (formatted in the journal style).

      I did not withdraw any reasoning. Since you do not understand it "as is", I offered some details. You claim that you still do not understand. I am sorry. If you do not understand why a mathematician would be unable to write such things in a research paper, you have no idea what the mathematics is. Your credentials, if any, are irrelevant here. Sir Timothy Gowers also hardly understands this.

      You refuse to tell your motivation. Then why should I tell you my? I stated an opinion, not a theorem.

      But I would like to correct you: I was interested in the composition of the committee, and looked for an explanation of an anomaly in the composition. That phrase supports one of two most probable explanations, and I stumbled at it after I arrived at my conclusions. It is not the basis for my opinion.

      Also, I don't see any strong statements here. I am saying that some guy is not a mathematician, what's the problem?

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    29. To make the links work, just copy them into browser line. The papers are in Inventiones mathematicae and the Annals of Mathematics, the titles: "Instanton counting on blowup. I. 4-dimensional pure gauge theory", "Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems", "The universal relation between scaling exponents in first-passage percolation", "The connective constant of the honeycomb lattice equals sqrt{2+sqrt{2}}".

      All this papers (an many more) contain words like "rigorous proof" and/or "mathematical" in their abstracts, which are available from any computer connected to Internet. These examples leave you with only two honest options: either take a responsability to claim that their authors, including e. g. Sourav Chatterjee, Horng-Tzer Yau, Stanislav Smirnov, are not mathematicians, or admit that the following statements were wrong:
      "A (pure) mathematician would never write in a research paper that some statement is "mathematical"."
      "mathematician would be unable to write such things"
      "nobody stresses rigor."
      "If you do not understand why a mathematician would be unable to write such things in a research paper, you have no idea what the mathematics is."

      Of course, you have a third option to discuss instead the evil Timothy Gowers, chemist teachers and my credentials (if any), but I wouldn't call it honest.

      Answering to your last question, if you insist: dubbing someone a "non-mathematician" mantra is obviously meaningless, but not at all harmless; it is widely used to eliminate a strong candidate from mathematical physics (and some other areas) during hiring procedures. Also, if you were not such an extremely knowledgable person as you are, your unequivocal statements and my counterexamples would indicate that you are completely unfamiliar with pure mathematical writing relevant for physics. This is unlikely; the other explanation I see is a "blind spot" caused by prejudice towards Hairer. In either case, your past or future judgements about him are obvously of little value. Prejudice, of course, is also the most common cause to use the mantra in general.

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    30. mtsyr: Wow, I had no idea that URL can be copy-pasted in the browser address. That's unbelievable. You know, with this advise you crossed a line. I left livejournal.com to a big extent because of such manners of many users. You are not welcome here anymore. I hope that the technological means to deal with this situation will not be needed.

      After following the first "link" I got an offer to buy the paper for $44.95. If I will do some extra manipulations, I will be able to download it for free, but, as I told you, I am too lazy to do this. The authors of the paper in Inventiones are mathematicians, and at least one of them is a much better than Hairer no matter how he and Hairer are labeled. The use of the words "mathematically rigorous proof" in this paper is well justified by the context and recent history. All relevant issues were discussed and settled in early 1990-ies.

      This is my blog, and I will discuss here whatever I like. Very often I do not have time to respond to meaningful and polite requests. I am not interested at all in your credentials. I am much more concerned about hiring chemists, than by non-hiring of mathematical physicists of all sorts.

      Your accusations are pointless. I admitted from the very beginning that I know nothing about works of Hairer and I would prefer to defer any judgement till the time when some detailed outline of his work will be available. I believe that this would be a reasonable and honest approach. In contrast with Sir Th. Gowers, who was a member of Fields medal committee in 2010 and had no idea about work of the medalists, I do not award any prizes and do not distribute any funds.

      Unfortunately, first Jamie Vitara, and now you forced me to look into Hairer's work and to form a premature opinion. The more pressure like this will be applied, the more negative my opinion about Hairer is going to be. There is only one way to change my opinion in other direction: to explain his work in a language accessible to non-experts. If I will be impressed, I will change my opinion.

      And I have no problems with admitting my "prejudice". I learned this hard way: it is much safer to presume anybody who "looks like" an applied mathematician to be a crook. There are good reasons for this, but this is not the topic I would like to discuss now.

      Have a nice life.

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    31. Dear Owl, I apologize for my tone. It is easy to lose temper when one provides something clearly relevant for the discussion and hears back what sounds like "I'm too lazy to follow the link"; I did not notice that you were talking about the full texts, as my point is already clear from the abstracts. That said, I admit that I should have put my comment in a more acceptable form.

      I was not, in fact, trying to "push" your opinion about Hairer in any direction. I have no idea about the level of his work, or what this "level" means, even less about whether there were better candidates. I was surprised by one concrete point of your reasoning, namely, dubbing someone "not a mathematician" since a mathematician would "never" do a certain thing. Now you seem to agree that this "never" suffers exceptions.

      Anyway, the main point of this comment was to apologize, and unless you are willing to accept the apologies, I will not bother you any longer. Thank you for interesting conversation.

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    32. A good way to get some idea about Hairer's work is to watch his talk at ICM with an introduction by Zeitouni. Another talk by Borodin also mentions Hairer's work and puts it in some perspective. Actually, if you have an hour to waste, please, watch Borodin's talk. He is a great speaker and gives a good motivation, including for Hairer's work.

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    33. mtsyr: Apologies are accepted. Of course, in order to say something about a paper, I need the whole paper, not just the abstract. Very often abstracts are nearly complete nonsense, especially when a journal requires an abstract.

      A couple of remarks about links - this may be useful in the future. First, these were not links, just addresses. Second, I do not follows any links unless I have some idea what awaits me there. You cannot post a link or an URL which will give me access to a paper behind a paywall. But you can post names of the authors, title, and year of the publications. It may happen that I know the paper, or I know the authors, what kind of mathematics they are doing, etc. And, in any case, these data will allow me to follow a routine process leading to the full access to the paper, if our university has a subscription to the journal.

      My opinion is not based on this argument alone. It was just a "cherry on the cake", a little nice addition.

      One should distinguish between a mathematical discussion and a discussion about mathematics. In mathematics "never" means "never", like in "the unit ball in an infinite dimensional Banach space is never compact". But when the discussion about mathematics, it is not much different from a discussion about French literature or Italian cities. Every universal statement has exceptions, unless it is completely trivial and not interesting at all. N. Bohr went much further: he said something like this: "Really interesting are only such true statements that they negation is also true".

      The claim that M. Mirzakhani is a mathematician is of no interest at all: this is obvious, and the negation is obviously wrong. M. Hairer is a much more interesting case: he is not a mathematician, and he is a mathematician. An attempt to understand what this main mean can be very instructive.

      You may try to thing about this. As of myself, I spend already more than enough time looking at Hairer's works, and my conclusion it that there is nothing interesting there. This was my presumption, then I found something potentially interesting, but eventually it turned out that the presumption was correct.

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    34. Correction: what this main mean --> what this may mean

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    35. Jamie Vitara: First, to listen to his own talk is useless for the purposes at hand. Nobody is expected to objectively evaluate her or his work in public. Just the opposite, everybody is expected to promote her or his own work.

      Second, I am unable to watch more than 10 minutes of a video recording of mathematical talk. After 10 minutes I am bored beyond all limits. The subject of the talk, the speaker, the venue are of no importance. The talk could be my personal friend about a highly interesting for me topic, the result is still the same. At the same time I can listen to a more than 2 hours long talk with full attention - in the audience. I don't know why, although I have some ideas. Basically, I blame the very low production quality, including the fact that mathematicians are usually very bad actors.

      As usual, there is an exception. There is a video recording of a talk by P. Halmos, which I watched entirely in two installments (still not the whole video at once!). P. Halmos was a brilliant speaker, and the production (by the AMS or MAA) was noticeably better than usual. It seems that Halmos took into account the fact that his talk will be recorded.

      Unfortunately, I don't have an hour to waste in this way. I have much better options.

      I will wait for written expositions. Actually, I will not wait. There are too many fascinating things around. If somebody else would get the 4-th medal, I would never knew that Hairer exists. "Applied mathematicians" badly need these toys: awards, prizes. Without these toys they would not exist for the world outside of that "applied mathematics". They works almost never have the beauty and the appeal of pure mathematics. They works are almost never useful for any applications. Pure mathematics is much more useful.

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    36. Well, in this case the set exceptions is so waste that it qualiifies for a new rule. Like it or not, when mathematics intertwines with physics literature, obvious social and political issues arise. This, naturally, affects writing.

      I'm puzzled where does the perception of Hairer as "an applied mathematician" come from. There seem to be no claims of "applicability" of his results. True, the whole area was motivated by physics; however, my understanding is that KPZ universality class was primarily interesting not due to "engeneering applications", but as (the only?) source of interesting exaclty solvable models of nonequilibrium growth. I have no idea about "deepness" of Hairer's work, but I know that such models were studied by many first-class mathematicians, and all these contributions circumvened the physicist's original SPDE approach, in particular due to lack of satisfactory theory. If the results of Hairer will ever be "applied" (I have no idea how likely is that), e. g. to connect two approaches, then it is most likely to be an application of pure mathematics to pure mathematics.

      Aside, this raises an interesting question of whether mathematical work should be judged by "deepness" only, or should impact (in particularly, outside mathematics) be also taken into account.The latter approach is supported both by a long tradition (Newton is a perfect example) and by an obvious strong correlation to deepness, otherwise a very shaky notion. It is easy to claim "there is nothing interesting in the work" (I can say that, being familiar with some other comments by Sowa, I guessed this conclusion from the beginning), but it is harder to argue e. g. with "it is an important field with a lot of good people who all missed this". Perhaps, it is better to discuss the last question (if any) with some other examples than Hairer.

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    37. mtsyr: The cardinality of the set of exceptions is irrelevant. If we will start discussing mathematics taking into account how many people do this or that, there will be nothing to discuss. 99.9999% of the people don't even know that mathematics exists in any other form than something they teach in schools. Much more people are working in physics than in mathematics. They always be a majority. In fact, there are much more mediocre mathematicians than good ones. If we will listen to their opinion, then - I don't know, I will just quit mathematics.

      Newton is not an example. Newton was not interested in any sort of impact. He wasn't even interested in publishing his results. Newton's work is very deep as pure mathematics irrespectively of any applications. And his "applications" to celestial mechanics were used for the first time after about a hundred of years passed, and these "applications" were to problems as useless as any pure mathematical problem.

      There is a strong inverse correlation between depth and "impact" outside of mathematics.

      And in what sense a mathematical work should be "judged"? Are we, pure mathematicians, committing some crime, and can be pardoned if we present an excuse in the form of some "impact"?

      Mathematics is an art. The impact of a novel is never comparable with the impact of an article in New York Times. Still, novels are sometimes deep, beautiful, etc. New York Times has no such qualities and do not strives to have them. It strives to have impact. The novels are a form of art. The best ones are understood only by few people.

      If you are interested in the outside "importance", you are at the wrong place. I am interested only in the internal for mathematics characteristics. Using any external criteria for any purposes distorts the subject. Fields medals used to be awarded only on the basis of internal criteria. Not anymore. This is sad. But this blog was started with quotes from Andre Weil about fragility of the art of mathematics, existing by an accident only. It may disappear completely.

      As of Hairer, the discussion is very strange. You have no idea about the level of his achievements, but you continue to defend him. I have reasons to be suspicious which I partially explained. But I have no chance to convince you, because you don't know Hairer's work, and you don't know the work of real mathematicians bypassed by the Fields committee. You believe that "someone is wrong on the internet", but you even don't have any positive arguments and are trying to find flaws in my comments. There are a lot of flaws here, I do not write comments with such care as I write proofs.

      But I am a little bit tired of playing the game "someone is wrong on the internet". You are wrong, but why should I attempt to convince you?

      I should write another post "What is mathematics?" The lack of understanding of what mathematics is leads to many problems, and to many mishaps, like awarding this medal to Hairer.

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  8. Don't you consider Quillen to be a "theory-builder" among the Fields Medalists?

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  9. shopkins: I do.

    Many people claim that J. Lurie has not specific results at all, only a big theory. He did "solved a problem", but this not an old problem and was hardly known before he solved it (in fact, this is often the case at the highest levels of achievement). These people dismiss this solution as not interesting at all.

    By this reason I picked up two extreme cases of L. Schwartz and A. Grothendieck.

    Many mathematicians awarded Fields medal in 1936-1990 are theory-builders. But most of them also solved problems or at least got some very specific results which were always considered as very important.

    D. Quillen proved the so-called Adams conjecture, which was considered during the 1960-ies the most important problem in topology. His creation of higher algebraic K-theory (pure theory-building) resulted from his ideas about the Adams conjecture. Later he proved, independently and simultaneously with A. Suslin the Serre's conjecture about projective modules over polynomial rings, which was also one of the central problems at the time.

    D. Quillen is one of my mathematical heroes, I count him among four-five best mathematicians of the previous century, who surpass all others by a wide margin.

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  10. Replies
    1. Vlad Nigmatullin: Let it will be four. Actually, I mean not the calendar century - this division is too arbitrary - but some sort "after-Hilbert" century. It is not ended yet.

      A. Weil, A. Grothendieck, D. Quillen, W. Thurston

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    2. You don't think Serre belongs on that list or Deligne.

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    3. Alex Reed: I would hate to say anything negative about J.-P. Serre or P. Deligne, even of the sort "he is not as great as Newton". So, I will limit myself by two remarks.

      J.-P. Serre worked mostly in the same area as A. Grothendieck. Both have Ph.D. in other areas (Serre in algebraic topology - this is the work for which he was awarded a Fields medal; Grothendieck in functional analysis), and both worked in algebraic geometry afterwards. P. Deligne is a pupil of Grothendieck and all his work is algebraic geometry. There is no doubt that people working in algebraic geometry and related areas consider Grothendieck as a genius beyond comprehension, "...summoned from the void".

      And, of course, this is *my* list. I do not hand out any positions or funds. Think of this list as a list of "Owl prize winners". The winners are not even announced, when they do win it. :-) I am not accepting any responsibility for my choice. I am not the IMU and I do not represent anybody but myself.

      I would be much more willing to explain why somebody *is* on the list, but this question wasn't asked. I hope that this is not an omission, that it is obvious that these four should be included. Actually, there is one more guy, whom I did not included because his style is closest to the one I always desired to follow (even before I learned about it) - I am not sure *myself* that in his case I am objective.

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    4. Thank You for the clarification. The reason I asked the question was because of your statement that these mathematicians "surpass all others by a wide margin". I think an argument can be made that Serre belongs in the same category as Quillen or perhaps surpasses him, He did after all make great contributions in Algebraic Topology, Algebraic Geometry and Algebraic Number Theory. Perhaps you might explain why Quillen is on your list when Serre is not. As for the rest of the list I agree with you.

      Just out of curiosity, who is the mathematician you referred to about whom you don't consider yourself objective.

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    5. Alex Reed: First, I am really not inclined to name the fifth person here. Actually, you questions and my attempts to answer them are leading me to the conclusion that there is no such person, and my idea resulted from an accident. There are other mathematicians of about the same level whose style of work is similar.

      I was sure that if I will be asked about why somebody is included, then I will be asked about Quillen for sure. As I said, I cannot say anything negative about J.-P. Serre. There is a difference between them which, may be, explains why I expected a question about Quillen. J.-P. Serre is an excellent mathematical writer. He wrote a lot of detailed paper, several books at different levels, his collected papers fill in 4 big volumes plus some extra publications (his talks not included in his collected papers). Quillen wrote very dense papers, many of which are very difficult to read because they are very condensed. Some of his papers were written by other people on the basis of his lectures, some are not published even if a manuscript exists.

      All main Quillen’s ideas have this remarkable almost self-contradictory quality of being totally unexpected and inevitable. A good example is his definition higher algebraic K-functors. To find a definition of a reasonable analogue of K_0, K_1, and later K_2 functors was one of the central problems in the 1960-ies (by the way, this problem is a good illustration of the difficulty of using the word "problem" - this is not a "problem" in the traditional sense). Eventually, several mathematicians nearly simultaneously suggested several definitions. All of them, except Quillen's, attempted to emulate some well known constructions in a different setting. All of them were more or less just definitions - there were no tools to work with them. Quillen's first definition, his famous plus-construction, was completely different and it was working. Very soon he suggested even better definition. The latter work was cited as the main reason for awarding a Fields medal to him. Both his definitions were immediately accepted as the "right" ones. The other definitions were soon proved to be equivalent or nearly equivalent to his, but it is possible to use them only because they are equivalent to Quillen's definition.

      A similar thing happened earlier with Quillen's work on the so-called Adams conjecture, a rather esoteric conjecture in algebraic topology, which was deservedly considered as one of the main problems in algebraic topology. The best topologists were not able even to approach it till Quillen suggested to use Grothendieck's algebraic geometry. He published a proof with a specific technical statement, rather plausible, left unproven. D. Sullivan modified his idea and found a proof. In the meantime Quillen used the theory of modular representation of finite groups in order to provide an independent complete proof. His second proof led him to his first definition of higher algebraic K-functors. The first proof was completed by his student E. Friedlander. The very idea of using a theory hidden deeply inside of algebra (even now not many people know that it exists) is striking. The creation of a new branch of mathematics, the higher algebraic K-theory as a result is even more striking.

      Quillen's ideas appear to be "summoned from the void", like Grothendieck’s ones. Note that Grothendieck also had competitors and people working in parallel - the first proof of one of the Weil conjectures is due not to Grothendieck, but to B. Dwork.


      Continued in the next comment.

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    6. There is a completely different quality of Quillen's work, which sets him aside. At the time he was the only person who saw topology and algebra as the same subject. While algebraic topology was designed to reduce topological problems to algebraic ones, Quillen freely moved between two fields, and his solution of, apparently, pure algebraic problem of defining higher K-functors was based on tools from topology. Grothendieck and before him A. Weil come close to the realization that this is indeed one subject, but not quite - both wanted to emulate topological theories within not even algebra, but algebraic geometry (which is, of course, a sort of geometry). This idea is still not internalized by the mathematical community. It seems that Lurie works like algebra and topology are the same branch of mathematics, but he is an exception. If you will read any textbook of algebraic topology, you will be read pure algebra presented in such a way that it looks like geometry. This is confusing, of course.

      Later in his career, after the Fields medal, Quillen suddenly moves to a new area and in an about 5 pages long paper creates a new field: the local index theorems. Unfortunately, the development of this theory was done by people working a completely different style, and the result is fairly scary.

      Well, may be this at least partially explains why I consider Quillen to be truly exceptional.

      Finally, I feel that all four share the same quality: they were so far ahead of time, that there was no adequate language to communicate their ideas (there is still no such language). The problem is very deep: the linear time-like structure of our communication is not adequate. Quillen attempted to deal with this by condensing his papers almost beyond limits, Grothendieck - by relying on the simultaneous work of his students/collaborators, Thurston - by not writing his papers at all. Weil dealt with this problem in a more traditional way - by building broad foundations for his ideas, which were made obsolete by Grothendieck.

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    7. Please explain to me why algebra and topology are the same subject. Also, please elaborate further on why "the linear time-like structure of our communication is not adequate".

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    8. I don't think that this is a suitable place for a discussion of either of these two topics. These topic are independent, and asking about both of them in the same comment doesn't looks nice. Using for such comments an account created the same day does not look nice either (in livejournal this was hardly acceptable).

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  11. Hi, Owl! First I want to say that I like your blog very much and I wish you would write here more often.

    You talked about areas of mathematics, "fine divisions". I'm new to mathematics (1st year), and I would like to know your opinion about this:

    I was talking to a friend and I mentioned the "square peg problem" as intriguing, and told him that it was amazing that a "simple" problem in geometry was open for such a long time. Then he said to me that although the problem is in geometry, to solve this type of problem, mathematicians first need to settle an equivalent algebraic formulation, and then the problem becomes a problem in algebra.

    Is this true?

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  12. Noel Gardeli: I don't know what equivalent algebraic version your friend had in mind. But reduction of a geometric problem to an algebraic problem is a standard method, actually algebraic topology was designed to do this. Apparently, at least some approaches to this problem are very much in spirit of algebraic topology. On the other hand, the problem is solved for "nice enough" curves, for example for curves defined by two times differentiable functions, but not for just continuous curves. Usually this indicated that there is a non-algebraic difficulty.

    Anyhow, all good pure mathematics is interrelated, and the division of it into Algebra, Geometry, and Analysis was obsolete already more than 50 years ago, despite many people manage to stay in one of these branches forever.

    A warning: problems like the "square peg problem" are amusing, but one should not spend to much time on them. It is hard to expect that they lead anywhere.

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  13. congrats to Sowa for being so amazingly psychic about the awards. doubt that many mathematicians could have matched this prowess and its based on a very deep grasp of both math & politics. amused as always by the intense "inside baseball", the moderate hostility toward applied mathematics & Hairer in particular who also has a strong CS side to him. more opinions in Fields/Nevanlinna prizes 2014

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  14. Anyone know hat happened in 2002, when the committee elected just two Fields laureates instead of four?

    I believe that, at the time, Grisha Perelman was already eligible for the prize due to his achievements in solving long standing problems in geometry. It was necessary for the guy to solve one of hardest problems in history to be elected to the Fields. Shame on you, IMU.

    It raises some relevant questions: What is the level of influence of Politics in deciding the Fields medalists? The fact that IMU is dominated by UK and France has an influence in the process?

    Perelman was the last great problem solver in history. Much of it, I believe, was due to the Politics and misconduct in the research community. I agree with some people that the golden age of Math has gone...

    What is you opinion, guys?

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    1. edwardlyon.mitchell: There is nothing special in awarding only two medals. For the first time more than two medals (four) were awarded in 1996. In 1982 only 3 medals were awarded. There was a more than natural 4th candidate, but, apparently, a lot of politics intervened, from international relations to the friction between various branches of mathematics. The 1982 Congress was exceptional in many respects, starting with the fact that it was held in 1983, not in 1983 (the medalists were announced in August 1982). In 1986 everything was back to normal, and the medals went to three mathematician about whom nobody expressed any serious doubts. Starting with 1990, the role if both internal mathematical politics and of external politics started to increase.

      2002 is another exception. The influence of any sort of politics was minimal. As far as I know, the Committee was unwilling to award two medals in the same area of mathematics, and treated the notion of "the same area" fairly broadly (but still more narrow than the archaic division of mathematics into Algebra, Geometry, and Analysis). A natural candidate was excluded because he was 3 weeks older than the 40 years rule allowed. And the Committee decided that there are no more candidates deserving it. I was told that the Committee had to fight with the IMU, which strongly preferred awarding 4 medals, not 2.

      I disagree about Grisha Perelman. I suspect that knowing the fact that by 2002 Grisha nearly completed his work on Poincare conjecture is distorting your perspective. Between 1995 and 2002 Grisha did not made public any new result, and only a handful of people (may be even just 2 or 3) knew that he is doing something. Secretly. That particular problem in riemannian geometry, which he solved in 1994 or 1995 wasn't the sort of problem which can ensure a Fields medal. It is an internal problem of riemannian geometry, hardly known even to most of differential geometers. It doesn't matter how you or me value this result. I just don't see any way to convince the mathematical community, even in an ideal world, that this is a Fields medal level work.

      And I don't see how awarding the medal to Grisha in 2002 could benefit either him or mathematics. He would reject it, as he already rejected the European Mathematical Union prize for the same work.

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    2. edwardlyon.mitchell: continuation.

      It seems that nowadays the choice of medalists is determined almost exclusively by politics.

      I am not aware of any evidence that IMU is dominated by UK and France. If it is dominated by any country or a small group of countries, then it is either by the US or by the US together with the West Europe.

      Grisha is not the last great problem solver in history. First of all, you should never say "never": you don't know the future. In any case, there are many problem solvers, much more than so-called "theory-builders". Who is deemed to be the best is determined by the current politics: which problem is considered to be important, which is not.

      Poincare himself did not considered his question to of such paramount importance. He did not stated the "Poincare conjecture", he just wrote that it would be interesting to know if it is true or not. He never considered questions with yes/no answer as important ones. He preferred "open questions". He had an obvious motivation to ask this particular question: he mistakenly believed that a much stronger statement (with the fundamental group replaced by homology) is true, and published a counterexample. He wanted to know how much new information is contained in the fundamental group compared to homology. This was - morally - sorted out by 1950 for sure. Of course, technical results about the relations of the fundamental group (or, rather, the homotopy groups, discovered in 1930ies) and the homology groups will never cease to appear (if mathematics survives).

      Technically, the Poincare conjecture is useless: if somebody is able to prove that the fundamental group of a 3-manifold is trivial, she or he is able to prove directly that the manifold is homeomorphic to the 3-sphere.

      The Thurston Geometrization Conjecture is incomparable more important. One can even argue that the original Thurston Geometrization Theorem is more important then more important than a proof of his conjecture, but this will lead as too far away. Anyhow, Grisha proved also the Thurston's Conjecture, and for me this is his main achievement, not the Poincare Conjecture.

      Golden Ages of anything do not last long. The Golden Ages of Mathematics return often enough to be assured in its ability to survive without some external intervention. Unfortunately, we now living in the age of a wide external intervention.

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  15. Any predictions for 2018? Who will win? Who should win? (if it was to be decided now) It's a pleasure to read your posts, you probably had fun writing them. ;)

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  16. J.S.:

    No, no predictions. I lost interest in this vanity fair.

    Since the very first Abel prize, I watched the announcements online. The last two times (2015 and 2016) I forgot about them completely and even did not bothered to find out the date. Both times I learned the winners from the AMS site. I have no problems with both choices. On the other hand, they are not interesting. Of course, A. Wiles is an exceptionally good mathematician – but we all knew this long ago.

    The Fields medals are just the opposite. There are about, say 80 remarkable mathematicians about the Fields medal age limit. 40 of them are somewhat over this limit and are excluded. Then a secret committee selects random 4 out of 40 and awards the medals. Other 36 are as good as these 4, and many of the 40 who are over the age limit are better. In fact, these 4 are selected not randomly (this would be better), but under pressure of various political groups.

    As I already said: would I knew the composition of the committee, I would be able if not to predict the winners, then to compile a list of 8 which will include all the winners. From 1978 till 1990 I was able to do this without knowing anything about the composition of the committee. From 1992 till 2006 I was able to guess correctly 2-3 out of 4 (out of 2 in 2002).

    I think that closer to the next congress I will know, or would be able to know enough about the political winds to made a reasonably good prediction.

    The problem is: would I be still interested in this game?

    If I will, at least a little bit, I will write about. For sure.

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  17. I think unfair to give the prize for analysis specialists, because, for example , how to assess Avila working on an individual basis ? In your case , 2 to 3 employees per paper. you can not see what he imagined.
    Or am I wrong?

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  18. Prime Goat:

    Sorry for the belated reply.

    This phenomenon is not specific to analysis. It seems that papers with only 1 author are disappearing.

    For one things, at sometimes it known who contributed the main ideas. Sometimes it is clear that all papers where X is one of the authors are of highest level, and the papers of her/his coauthors without X are not. Sometimes the experts know much more than the general mathematical public.

    But the real problem is not here. The problem is a sort of "cult of personality" in mathematics. Mathematicians award prizes to persons, not to the results, in contrast with, say, Nobel prizes.

    It would be much better to award prizes to results (and to consider a theory as a result). There would be no questions about the individual contributions. And the prizes would be much better in telling the community what is good/important mathematics – which is, it seems, the only way prizes can be useful. Fields medals are not useful in this way anymore.

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