Previous post: 2014 Fields medalists?
This was a relatively easy task during about three decades. But it is nearly impossible now, at least if you do not belong to the “inner circle” of the current President of the International Mathematical Union. But they change at each Congress, and one can hardly hope to belong to the inner circle of all of them.
I would like to try to explain my approach to judging a particular selection of Fields medalists and to fairly efficiently guessing the winners in the past. This cannot be done without going a little bit into the history of Fields medals as it appears to a mathematician and not to a historian working with archives. I have no idea how to get to the relevant archives and even if they exist. I suspect that there is no written record of the deliberations of any Fields medal committee.
The first two Fields medals were awarded in 1936 to Lars Ahlfors and Jesse Douglas. It was the first award, and it wasn’t a big deal. It looks like that the man behind this choice was Constantin Carathéodory. I think that this was a very good choice. In my personal opinion, Lars Ahlfors is the best analyst of the previous century, and he did his most important work after the award, which is important in view of the terms of the Fields’ will. Actually, his best work was done after WWII. If not the war, it would be done earlier, but still after the award. J. Douglas solved the main problem about minimal surfaces (in the usual 3-dimensional space) at the time. He did with the bare hands things that we do now using powerful frameworks developed later. I believe that he became seriously ill soon afterward, but today I failed to find online any confirmation of this. Now I remember that I was just told about his illness. Apparently, he did not produce any significant results later. Would he continue to work on minimal surfaces, he could be forced to develop at least some of later tools.
The next two Fields medals were awarded in 1950 and since 1950 from 2 to 4 medals were awarded every 4 years. Initially the International Mathematical Union (abbreviated as IMU) was able to fund only 2 medals (despite the fact that the monetary part is negligible), but already for several decades it has enough funds for 4 medals (the direct monetary value remains to be negligible). I was told that awarding only 2 medals in 2002 turned out to be possible only after a long battle between the Committee (or rather its Chair, S.P. Novikov) and the officials of the IMU. So, I am not alone in thinking that sometimes there are no good enough candidates for 4 medals.
I apply to the current candidates the standard of golden years of both mathematics and the Fields medals. For mathematics, they are approximately 1940-1980, with some predecessors earlier and some spill-overs later. For medals, they are 1936-1986 with some spill-overs later. The whole history of the Fields medals can traced in the Proceedings of Congresses. They are interesting in many other respects too. For example, they contain a lot of very good expository papers (and many more of bad ones). It is worthwhile at least to browse them. Now they are freely available online: ICM Proceedings 1893-2010.
The presentation of work of 1954 medalists J.-P. Serre and K. Kodaira by H. Weyl is a pleasure to read. H. Weyl unequivocally tells that their mathematics is new and went into a new territory and is based on methods unknown to most of mathematicians at the time (in fact, this is still true). He even included an introduction to these methods in the published version.
The 1990 award at the Kyoto Congress was a turning point. Ludwig D. Faddeev was the Chairman of the Fields Medal Committee and the President of the IMU for the preceding 4 years. 3 out of 4 medals went to scientists significant part of whose works was directly related to his or his students’ works. The influence went in both directions: for one winner the influence went mostly from L.D. Faddeev and his pupils, for two other winners their work turned out to be very suitable for a synthesis with some ideas of L.D. Faddeev and his pupils. All these works are related to the theoretical physics. Actually, after reading the recollections of L.D. Faddeev and prefaces to his books, it is completely clear that he is a theoretical physicists at heart, despite he has some interesting mathematical results and he is formally (judging by the positions he held, for example) considered to be a mathematician.
The 1990 was the only year when one of the medals went to a physicist. Naturally, he never proved a theorem. But his papers from 1980-1994 contain a lot of mathematical content, mostly conjectures motivated by quantum field theory reasoning. There is no doubt that his ideas are highly original from the point of view of a mathematician (and much less so from the point of view of someone using Feynman’s integrals daily), that they provided mathematicians with a lot of problems to think about, and indeed resulted in quite interesting developments in mathematics. But many mathematicians, including myself, believe that the Fields medals should be awarded to outstanding mathematicians, and a mathematician should prove his or her claims. I don’t know any award in mathematics which could be awarded for conjectures only.
In 1994 one of the medals went to the son of the President of the IMU at the time. Many people think that this is far beyond any ethical norms. The President could resign from his position the moment the name of his son surfaced. Moreover, he should decline the offer of this position in 1990. It is impossible to believe that that guy did not suspect that his son will be a viable candidate in 2-3 years (if his son indeed deserved the medal). The President of IMU is the person who is able, if he or she wants, to essentially determine the winners, because the selection of the members of the Fields medal Committee is essentially in his or her hands (unless there is a insurrection in the community – but this never happened).
As a result, the system was completely destroyed in just two cycles without any changes in bylaws or procedures (since the procedures are kept in a secret, I cannot be sure about the latter). Still, some really good mathematicians got a medal. Moreover, in 2002 it looked like the system recovered. Unfortunately already in 2006 things were the same as in the 1990ies. One of the awards was outrageous on ethical grounds (completely different from 1994); the long negotiations with Grisha Perelman remind plays by Eugène Ionesco.
In the current situation I would be able to predict the winners if I would knew the composition of the committee. Since this is impossible, I will pretend that the committee is as impartial as it was in 1950-1986. This is almost (but not completely) equivalent to telling my preferences.
I would be especially happy if an impartial committee will award only 2 medals and Manjul Bhargava and Jacob Lurie will be the winners. I hope that their advisors are not on the committee. Their works look very attractive to me. I suspect that Jacob Lurie is the only mathematician working now and comparable with the giants of the golden age. But I do not have enough time to study his papers, or, rather, his books. They are just too long for everybody except people working in the same field. Usually they are hundreds pages long; his only published book (which covers only preliminaries) is almost 1000 pages long. Papers by Manjul Bhargava seem to be more accessible (definitely, they are much shorter). But I am not an expert in his field and I would need to study a lot before jumping into his papers. I do not have enough motivation for this now. An impartial committee would be reinforce my high opinion about their work and provide an additional stimulus to study them deeper. The problem is that I have no reason to expect the committee to be impartial.
Arthur Avila is very strong, or so tell me my expert friends. His field is too narrow for my taste. The main problem is that his case is bound to be political. It depends on the balance of power between, approximately, Cambridge, MA – Berkley and Rio de Janeiro – Paris. Here I had intentionally distorted the geolocation data.
The high ratings in that poll of Manjul Bhargava and Artur Avila are the examples of the “name recognition” I mentioned. I think that an article about Manjul Bhargava appeared even in the New York Times. Being a strong mathematician from a so-called developing country (it seems that the term “non-declining” would be better for English-speaking countries), Artur Avila is known much better than American or British mathematicians of the same level.
Most of mathematicians included in the poll wouldn’t be ever considered by anybody as candidates during the golden age. There would be several dozens of the same level in the same broadly defined area of mathematical. Sections of the Congress can serve as the first approximation to a good notion of an area of mathematics. And a Fields medalist was supposed to be really outstanding. Restricting myself by the poll list I prefer one of the following variants: either Bhargava, or Lurie, or both or no medals for the lack of suitable candidates.
Next post: Did J. Lurie solved any big problem?
Thank you for writing such an interesting post, Sowa! I am a mathematician at the beginning of my career, and this historical perspective is very hard to find. I think Arnold wrote somewhere that in 1994 three of the winners were noted "for the art of manipulation of inequalities".
ReplyDeleteHere are some link for people to connect the dots in your story: Fields Medalists, Former Prize Selection Committees, Former Executive Committees.
I will comment more about your post a little later.
The award stipulation is vague - in recognition of work already done, while at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others. Because of the diversity of mathematics, it impossible to make the rule precise. There is this popular division into theory builders and problem solvers, but I think there are many shades in between. Maybe, the question should be - what impresses us in other people's work? The most common reason (because it is the easiest to appreciate psychologically) is when someone solves a long standing famous open problem, or make impressive progress towards it. On the opposite extreme, is when someone develops an expansive theory that seems completely new and mysterious; it takes a lot of time to understand. As you say, sometimes even 50 years later people don't completely understand it. Then there are various cases in between. For example, someone develops a few new ideas that solve several (maybe not spectacular but very difficult) open problems. Or someone masterfully combines ideas from different fields that solve several (again, maybe not spectacular but difficult) open problems. Or, perhaps, there is a real obstacle to progress in some area, and removing it opens the floodgates so to speak. I think we should have an open mind and, in this sense, it is probably not fair to compare only to the giants of the golden age, as you say, because even among themselves some are more "giant" than others.
ReplyDeleteI am glad you brought up Jacob Lurie because his work falls into the (higher?) category of completely new and mysterious. People said that he would win already in 2010, so I would be very surprised if he does not win in 2014. For educational purposes, can you give one or two examples of Jacob Lurie's work which are considered his deepest and most spectacular contributions?
In fact, mathematicians only rarely interested in the history of mathematics (apparently, this is true for most of other professions also). This is rather disappointing, because the true motivation for everything done in pure mathematics is the history of the question, and not the artificial motivations included in the textbooks (or posted on T. Gowers’s website). This very negatively affects even teaching, especially the low level teaching: students have to learn a lot of difficult and completely unmotivated ideas. The recent history, including the “administrative” history of the sort discussed in my post, is crucial for understanding the current situation, especially for distinguishing the current fade from fundamental trends.
ReplyDeleteThis is especially important for beginning mathematicians. Usually they just follow the current fade, but such a strategy will not lead you very far. As the late V.I. Arnold stressed several times in his publications, one should not follow the current, but apply a force directed orthogonally to the current. He used the following analogy. Suppose you are near the border of a fast rotating disc. In order to stay on disc you have to apply a force orthogonal to the direction of your motion, otherwise you will be thrown away from the disc very soon. This is similar to the satellites orbiting the Earth: the force keeping them on the orbit is the gravitation attraction to the Earth, which is orthogonal to the direction of the motion. Another relevant Arnold’s remark is the following: one should work on problems which will be fashionable 20 years from now, and not on the problems which are fashionable today.
I am very glad that you are interested in at least some aspects of the history. I must confess that I intentionally omitted a lot of names. Most of them can be reconstructed very easily, and wanted to stimulate the readers to look for them: in this way they will get not just a recognizable name, but at least some information about the person. In particular, all names (except the ones I cannot divulge, like the names of my expert friends) can be found in the Proceedings of Congresses in opening addresses.
I would like to warn against using Wikipedia. In many areas Wikipedia reached the level of a reliable source, and when the list of references is up to the Wikipedia standards, everything is not very hard to verify by original sources. This is far from being the case in mathematics: Wikipedia gives a very distorted and incomplete picture, and I still doubt that it ever be reliable. Well, edited by T. Gowers “The Princeton Companion to Mathematics” gives a picture no less distorted, but with an incomparable more substantial stamp of approval.
Yes, Arnold wrote this either in “The Mathematical Intelligencer”, or in the “Notices of the AMS”.
ReplyDeletePerhaps, the most accessible introduction to a part of Lurie’s work is a recent survey Daniel S. Freed, The Cobordism hypothesis, Bull. AMS, V. 50 (2013), 57-92 and his papers cited there. There is also a note by Lurie in the Notices of the AMS (a “What is ...?” note). It seems that the various versions of the notion of a higher category (especially of what he calls an (∞, 1)-category) are at the heart of his work. So, in order to understand his ideas, one should be as comfortable with categorical ideas as with natural numbers, and to realize that the higher categories are inevitable and to internalize this idea. I don’t have any recommendations about how to reach this stage. Examples of situations where they give a technical advantage may help, but they are not the raison d'être for higher categories. This question inevitably leads to the topic about which I still hope to write a couple of posts: why “abstract” proofs are simultaneously simpler and more difficult and what this really means? Why replacing 10 pages long elementary proof by a half-page long application of a theory requiring 500 pages for an exposition could be a tremendous progress?
Most of the questions raised in your second comment are very interesting and important. I will try to write a post about these issues. By the way, as you apparently already noticed, the prizes distort the picture, almost inevitably encouraging solutions of specific problems much more than building of powerful theories. Fields medals provide a good example.
I am surprised you recommend the Freed paper which seems to me very poor. It promises much but leaves the reader with no precise or useable idea.
DeleteDear Robert Walters,
DeleteIt even may be the case that all expository papers by D.S. Freed are very poor. Frankly, I did not read any of them. But this one is on my reading list. It seems to me that this one is fairly understandable and fairly short (especially compared to Lurie's own texts). I suggested it as an accessible introduction and a place where one may find further references. It is definitely accessible in the sense that it is available on the internet in its final form free of charge (this important for many people).
If you will suggest something else, I will be more than happy. An extra exposition (of a recent work) never hurts. It seems that there is no Seminaire Bourbaki talk about Lurie work, if I not missed it somehow. Personally, I would like to start with such a talk. Of course, some of them are terribly written (in such a case I will drop it), but some are excellent.
Dear Sowa,
DeleteI heard a criticism of Lurie's work, that it does not contain startling new ideas, complete solutions of important problems, even new conjectures. That he is simply rewriting old ideas in a new language. I am very far from this area, and I find it a little disturbing that only the ultimate experts speak highly of his work. Even people in related areas can not usually give specific examples of his greatness. I understand that his objectives may be much more long-term, but I would still like to hear some response to these criticisms.
Dear Tamas Gabal,
DeleteMy reply is in the next post Did J. Lurie solved any big problem?.
"Why replacing 10 pages long elementary proof by a half-page long application of a theory requiring 500 pages for an exposition could be a tremendous progress?"
ReplyDeleteThis statement is extremely fascinating because it is so counterintuitive. What if a ten page proof is the only proof we have and, in addition to being elementary, it is built on very new and beautiful ideas? To my taste it would be a perfect scenario and I can not imagine how the field might develop so that your statement becomes true?
I look forward to your posts on this and other issues. Your blog must have many readers, even if they do not always leave comments.
Maybe this is not so hard to understand. Ten pages of elementary proof might be a clear explanation of why some statement is true (or it might not, ten pages of calculation with matrices plucked from thin air can prove some types of theorem but still leave the reader no understanding of why the statement holds). But it is not too likely to be general, at least not at anything like the sort of level of generality someone like Lurie wants.
DeleteIn order to prove the more general result, one needs to find the right framework to place the basic result in, a framework in which the generalisation is natural. And that framework might be hard to understand, or require a lot of new tools to work with (think carefully about even simple classic proofs in number theory and you will find a lot of statements about the natural numbers being used without comment, whose generalisations require thought and proof). If you are the first person writing about the general framework, you need to do all this setting up, and it takes a lot of work. At the end, you have some powerful hammers which prove your basic result as a special case of the general result easily.
The hope is of course that your hammers can be used to generalise not just one result, but the whole theory, that the theory so generalised contains new and useful results (useful outside the theory) which weren't accessible in the basic theory, and that the proofs become easier in the general context (in some sense this is not possible, the basic proof will always be simpler - but the best attempt at generalisation in the basic theory may get very hard to prove, sowa's example running up to differential forms is a good example).
If the hope is satisfied, of course it is obvious why this is a great advance. You work hard to get the framework for the first result, but the second, third, hundredth are simplified and you are proving results you couldn't formulate in any understandable way in the basic theory. If the hope is not satisfied, and your big theory lies unused - then you wasted 500 1/2 pages on a statement that could be proved in ten pages.
Maybe this is not so hard to understand. Ten pages of elementary proof might be a clear explanation of why some statement is true (or it might not, ten pages of calculation with matrices plucked from thin air can prove some types of theorem but still leave the reader no understanding of why the statement holds). But it is not too likely to be general, at least not at anything like the sort of level of generality someone like Lurie wants.
DeleteIn order to prove the more general result, one needs to find the right framework to place the basic result in, a framework in which the generalisation is natural. And that framework might be hard to understand, or require a lot of new tools to work with (think carefully about even simple classic proofs in number theory and you will find a lot of statements about the natural numbers being used without comment, whose generalisations require thought and proof). If you are the first person writing about the general framework, you need to do all this setting up, and it takes a lot of work. At the end, you have some powerful hammers which prove your basic result as a special case of the general result easily.
The hope is of course that your hammers can be used to generalise not just one result, but the whole theory, that the theory so generalised contains new and useful results (useful outside the theory) which weren't accessible in the basic theory, and that the proofs become easier in the general context (in some sense this is not possible, the basic proof will always be simpler - but the best attempt at generalisation in the basic theory may get very hard to prove, sowa's example running up to differential forms is a good example).
If the hope is satisfied, of course it is obvious why this is a great advance. You work hard to get the framework for the first result, but the second, third, hundredth are simplified and you are proving results you couldn't formulate in any understandable way in the basic theory. If the hope is not satisfied, and your big theory lies unused - then you wasted 500 1/2 pages on a statement that could be proved in ten pages.
Dear Pete,
DeleteThanks for a great comment!
I still would like to give some specific examples. My problem is that I am not sure what counts as an example. Ideally, it would be an important result with a short and reasonably elementary, but not conceptual proof, which stimulated a successful search for a conceptual proof, no matter how long it is. But it seems to me that such things never happen: a short elementary proof not only discourages, but effectively prevents all work in the direction of the already proved theorem. Even parallel projects already in progress will be abandoned.
Apparently, I have to be satisfied by something less than this. For example, the conceptual proof may emerge from an independent development of an initially unrelated theory (at least not motivated by the result in question). Alternative, an elementary proof may be discovered after a (more) conceptual one, and turn out to be useless. It seems that the elementary proof of the Prime Numbers Theorem (about the distribution of primes) is a very good example of the latter situation.
Various proofs of the Brouwer fixed points theorem may serve as a sort of example in the original direction. Brouwer's own proof is, apparently, lies halfway between elementary and conceptual. Sperner's proof is completely elementary and even found its way in more or less popular books (I have in mind, first of all, Aigner-Ziegler). It was found later than the Brouwer's one, but, it seems, wasn't influenced by it. Finally, the right proof is the (co)homology theory proof. Sperner's proof requires almost nothing, the homology theory requires to study the homology theory first (about 200 pages for an exposition on the usual graduate level; may be even more). But the latter has extremely far reaching generalizations (say, Weil Conjectures) in contrast with Sperner’s one.
To add to my previous comment, an example that comes to mind is Paul Cohen's proof of the independence of the continuum hypothesis and the axiom of choice from the Zermelo–Fraenkel set theory. His original proof was perhaps 15-20 pages long and (please, correct me if I am wrong here) it is considered to be a perfect proof and has never really been improved. There must be more examples like this.
ReplyDeleteRight now I will comment only about P. Cohen’s theory, and only about the mentioned aspect of it. The first publication of Cohen’s results consisted of two 6 pages long notes in Proc. Nat. Acad. Sci. USA, a journal which never was intended for complete expositions of mathematical results. I never looked at these two notes; definitely, K. Gödel would be able to understand the proof from them (he did this with a preliminary version in just one day), but how many others would be able to do this? The detailed exposition was published by Cohen as a 192 pages long book; it was recently reprinted by Dover. It is true that the book contains not only his theorems. It begins with very condensed expositions of the mathematical logic and of the axiomatic set theory. These two chapters are great for somebody already familiar with these theories, but are hardly comprehensible for others. Apparently, Cohen needed to present the basics of these theories from his own point of view in order to be able to present the details of his proof. His own results occupy about 1/3 of his book, which is written in a very dense manner.
ReplyDeleteSo, first of all, Cohen’s proof is not very short. It is not elementary in the sense that it is based on essentially all tools from the mathematical logic and the axiomatic set theory available at the time. Second, Cohen not just proved a couple of great theorems. He created a theory, which he himself applied only to the most famous problems. Other mathematicians applied it to other problems.
Cohen’s theory is not an illustration of my remark. But it is an illustration of what type of work in mathematics is valued the most: the solution of an outstanding problem by creating a completely new theory changing the way we think about an area of mathematics. According to Hugh Woodin, by now Cohen’s forcing method is not just a tool to prove some theorems, it is the framework for all research in the axiomatic set theory.
Some people think that the theory of Boolean-valued models of Dana Scott provides a better way to prove Cohen’s theorem. Indeed, it has some advantages, but mathematically it is morally equivalent to Cohen’s theory. It is more or less a matter of taste which approach to use.
Well, perhaps I should say explicitly a couple of things I took for granted. Of course, I wanted to make a counterintuitive statement. I wanted to attract attention to some phenomena. Of course, nobody sets him/herself the goal of replacing a half-page proof by 500 pages long theory. Doesn’t make much sense, I agree. In your ideal imaginary example there is very little motivation to search for another proof. But already here there is a catch.
ReplyDeleteYu. I. Manin said in one (published) interview that he fears that Riemann Hypothesis will be proved (fairly soon) by “wrong methods”. What if somebody young and smart discovers a short proof of RH? A lot of other people will stop working on related questions, because deep down in their hearts all they wanted was to prove RH. The discovery of the “right proof” of RH will be delayed indefinitely, for centuries. But there is a widely accepted idea of what the right proof of RH should look like, and it will be a great disappointment to have only a wrong one.
So, we never set the goal of finding a long proof. The point is that many famous problems are hardly interesting per se. They are valuable only as a motivation to look for a proof in the hope that we will find the right one first. The Last Fermat Theorem is not interesting at all, in the opinion of Gauss, for example. We are lucky that it is proved as a corollary of other, really interesting results.
Some more explanations are still in order. The real challenge is to find good examples, because a “wrong” solution of a central problem in a field kills that field. The examples can be only indirect ones.
Ben Green will give a plenary lecture at ICM 2014. Considering an impressive list of past awards, he could be a strong candidate. By "strong" I mean in terms of likelihood. It does not hurt to be so closely linked to Gowers and Tao.
ReplyDeleteSowa, by the way, thank you for mentioning the plays of Eugene Ionesco. I read two and really enjoyed them.
ReplyDeleteJust to clarify, Manjul Bhargava is not from a developing country, he is from America.
ReplyDeleteHi!
DeleteThanks for commenting and paying attention to small details. Actually, I did not write that Manjul Bhargava is from a developing country, but I could (and should) write that paragraph more clearly. I tend to write in a more or less German and a partially Russian style; this was noticed and discussed a little in the comments to the Gowers's blog (to the post about this year Abel prize).
I meant that Avila is from Brazil, which is officially classified as a developing (and by an implication, not yet developed) country. It is indeed developing, but I am far from being sure that this has any relevance to awarding medals. Manjul Bhargava is, if I remember correctly, was born or at least got his school education in Canada. Canada is in America, but very often people interpret "America" as "USA". Even if he is born in the US, the American mentality will treat him both as an American, and as being from India, especially because he keeps his ties with the Indian culture (in contrast with T. Tao, for examples, who considers himself as Australian, or at least wrote so). So, actually Bhargava also has, to a lesser degree, an advantage as a guy from a "developing country".
Please, note that all this I suggested and/or suggest only as an explanation of ratings in that poll. Nothing related to citizenship, gender, ethnic origin, etc. should be taken into account even subconsciously by Fields Medal Committee. On the other hand, it is well known that US representatives regularly inject these aspects into all decision, in particularly in every question related to the Congress.
Amazingly, Ngo, 2010 Fields medalist, is listed as a Vietnamese mathematician. He was educated in France after finishing school, his advisor is French, and all his work is very French in the style, in the choice of problems, and in the methods. Considering him as being Vietnamese may serve some political goals, but it is obscuring our understanding of the "architecture of mathematics". The French school is very special.
I think you got Artur Avila's name wrong.
ReplyDeleteDear Xiao-Chuan Liu,
ReplyDeleteYour comment is very cryptic. What exactly I got wrong? Do you mean the mistake in spelling? Probably, I copy-pasted it from somewhere; anyhow I will correct it in a moment.