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
Showing posts with label Fields medals. Show all posts
Showing posts with label Fields medals. Show all posts
Wednesday, August 13, 2014
Tuesday, August 12, 2014
Who will get Fields medals in less than two hours?
Previous post: About expository writing: a reply to posic
At 10:30 p.m. US Eastern Summer time, the winner of this (2014) year Fields medals will be announced in Seoul.
I would like to post my current guess, mostly to have a record of it with the date and time stamp from Google, at least for myself.
As I wrote about one year ago, I believe that I would be able to predict the actual winners if I would know the composition of the Fields medals committee. But I don't. I am not particularly interested in the names of the winners, so I did not attempted to find out the actual winners, who are known for at least three months already, and who are known to the press for at least two weeks already (if the practice of the last two congresses was continued). So, my guess is a guess and not based on any inside sources.
And the winner are (expected to be):
Artur Avila - my confidence is over 95-99%.
One of the winners will be a woman - my confidence is over 95%. This is a pure politics. This deserves a separate discussion. The main obstruction to the Fields medal for a woman is not the discrimination, but the absurd age restriction. Most likely, she is
Sophie Morel - my confidence is over 80%. There are political consideration against her. For example, she would be the 3rd medalist who was a student of Gérard Laumon.
Jacob Lurie - my confidence is about 60%. This is my favorite candidate. He will get it if Harvard has enough political clout now. So, it is a measure of the influence of the Harvard Department of Mathematics, and not of the level of J. Lurie as a mathematician.
Manjul Bhargava - my confidence is less than 50%. If Sophie Morel gets a medal, his chances are much lower than otherwise: two mathematicians from the same university (Princeton).
Following the tradition firmly established since 1990, one of the medals should go a "Russian" mathematician, no matter where she or he is working know and where she or he completed Ph.D. I don't see any suitable candidate. Some people were naming Alexei Borodin, but I was firmly told that he will not get one.
A couple of days ago a strange, apparently unmotivated idea come to my mind: one of the winners will be from Stanford. Some people were naming Maryam Mirzakhani, but, again, a couple of days ago was firmly told that she is not the winner. Her work is interesting and close to my own interests. In my personal opinion, she has some very good results, but nothing of the Fields medal level. I would estimate the number of mathematician of about her level or higher, working in closely related areas, as at least 2-3 dozens. Of course, I am not aware about her most recent unpublished (at least on the web) work.
Next post: And who actually got Fields medals?
At 10:30 p.m. US Eastern Summer time, the winner of this (2014) year Fields medals will be announced in Seoul.
I would like to post my current guess, mostly to have a record of it with the date and time stamp from Google, at least for myself.
As I wrote about one year ago, I believe that I would be able to predict the actual winners if I would know the composition of the Fields medals committee. But I don't. I am not particularly interested in the names of the winners, so I did not attempted to find out the actual winners, who are known for at least three months already, and who are known to the press for at least two weeks already (if the practice of the last two congresses was continued). So, my guess is a guess and not based on any inside sources.
And the winner are (expected to be):
Artur Avila - my confidence is over 95-99%.
One of the winners will be a woman - my confidence is over 95%. This is a pure politics. This deserves a separate discussion. The main obstruction to the Fields medal for a woman is not the discrimination, but the absurd age restriction. Most likely, she is
Sophie Morel - my confidence is over 80%. There are political consideration against her. For example, she would be the 3rd medalist who was a student of Gérard Laumon.
Jacob Lurie - my confidence is about 60%. This is my favorite candidate. He will get it if Harvard has enough political clout now. So, it is a measure of the influence of the Harvard Department of Mathematics, and not of the level of J. Lurie as a mathematician.
Manjul Bhargava - my confidence is less than 50%. If Sophie Morel gets a medal, his chances are much lower than otherwise: two mathematicians from the same university (Princeton).
Following the tradition firmly established since 1990, one of the medals should go a "Russian" mathematician, no matter where she or he is working know and where she or he completed Ph.D. I don't see any suitable candidate. Some people were naming Alexei Borodin, but I was firmly told that he will not get one.
A couple of days ago a strange, apparently unmotivated idea come to my mind: one of the winners will be from Stanford. Some people were naming Maryam Mirzakhani, but, again, a couple of days ago was firmly told that she is not the winner. Her work is interesting and close to my own interests. In my personal opinion, she has some very good results, but nothing of the Fields medal level. I would estimate the number of mathematician of about her level or higher, working in closely related areas, as at least 2-3 dozens. Of course, I am not aware about her most recent unpublished (at least on the web) work.
Next post: And who actually got Fields medals?
Friday, August 23, 2013
The role of the problems
Previous post: Is algebraic geometry applied or pure mathematics?
From a comment by Tamas Gabal:
The obsession of modern pure mathematicians with famous problems is not quite healthy. The proper role of such problems is to serve as a testing ground for new ideas, concepts, and theories. The reasons for this obsession appear to be purely social and geopolitical. The mathematical Olympiads turned in a sort of professional sport, where the winner increases the prestige of their country. Fields medals, Clay’s millions, zillions of other prizes increase the social role of problem solving. The reason is obvious: a solution of a long standing problem is clearly an achievement. In contrast, a new theory may prove its significance in ten year (and this will disqualify its author for the Fields medal), but may prove this only after 50 years or even more, like Grassmann’s theory. By the way, this is the main difficulity in evaluating J. Lurie's work.
Poincaré wrote that problems with a “yes/no” answer are not really interesting. The vague problems of the type of explaining certain phenomena are the most interesting ones and most likely to lead to some genuinely new mathematics. In contrast with applied mathematics, an incremental progress is rare in the pure mathematics, and is not valued much. I am aware that many analysts will object (say, T. Tao in his initial incarnation as an expert in harmonic analysis), and may say that replacing 15/16 by 16/17 in some estimate (the fractions are invented by me on the spot) is a huge progress comparable with solving one of the Clay problems. Still, I hold a different opinion. With these fractions the goal is certainly to get the constant 1, and no matter how close to 1 you will get, you will still need a radically new idea to get 1.
It is interesting to note that mathematicians who selected the Clay problems were aware of the fact that “yes/no” answer is not always the desired one. They included into description of prize a clause to the effect that a counterexample (a “no” answer) for a conjecture included in the list does not automatically qualifies for the prize. The conjectures are such that a “yes” answer always qualifies, but a “no” answer is interesting only if it really clarifies the situation.
Next post: Graduate level textbooks I.
From a comment by Tamas Gabal:
“I also agree that many 'applied' areas of mathematics do not have famous open problems, unlike 'pure' areas. In 'applied' areas it is more difficult to make bold conjectures, because the questions are often imprecise. They are trying to explain certain phenomena and most efforts are devoted to incremental improvements of algorithms, estimates, etc.”
The obsession of modern pure mathematicians with famous problems is not quite healthy. The proper role of such problems is to serve as a testing ground for new ideas, concepts, and theories. The reasons for this obsession appear to be purely social and geopolitical. The mathematical Olympiads turned in a sort of professional sport, where the winner increases the prestige of their country. Fields medals, Clay’s millions, zillions of other prizes increase the social role of problem solving. The reason is obvious: a solution of a long standing problem is clearly an achievement. In contrast, a new theory may prove its significance in ten year (and this will disqualify its author for the Fields medal), but may prove this only after 50 years or even more, like Grassmann’s theory. By the way, this is the main difficulity in evaluating J. Lurie's work.
Poincaré wrote that problems with a “yes/no” answer are not really interesting. The vague problems of the type of explaining certain phenomena are the most interesting ones and most likely to lead to some genuinely new mathematics. In contrast with applied mathematics, an incremental progress is rare in the pure mathematics, and is not valued much. I am aware that many analysts will object (say, T. Tao in his initial incarnation as an expert in harmonic analysis), and may say that replacing 15/16 by 16/17 in some estimate (the fractions are invented by me on the spot) is a huge progress comparable with solving one of the Clay problems. Still, I hold a different opinion. With these fractions the goal is certainly to get the constant 1, and no matter how close to 1 you will get, you will still need a radically new idea to get 1.
It is interesting to note that mathematicians who selected the Clay problems were aware of the fact that “yes/no” answer is not always the desired one. They included into description of prize a clause to the effect that a counterexample (a “no” answer) for a conjecture included in the list does not automatically qualifies for the prize. The conjectures are such that a “yes” answer always qualifies, but a “no” answer is interesting only if it really clarifies the situation.
Next post: Graduate level textbooks I.
Monday, July 29, 2013
Guessing who will get Fields medals - Some history and 2014
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?
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?
Sunday, July 28, 2013
2014 Fields medalists?
Previous post: New comments to the post "What is mathematics?"
I was asked by Tamas Gabal about possible 2014 Fields medalists listed in an online poll. I am neither ready to systematically write down my thoughts about the prizes in general and Fields medals in particular, nor to predict who will get 2014 medals. I am sure that the world would be better without any prizes, especially without Fields medals. Also, in my opinion, no more than two persons deserve 2014 Fields medals. Instead of trying to argue these points, I will quote my reply to Tamas Gabal (slightly edited).
Somewhat later I wrote:
Tamas Gabal replied:
Here is my reply.
Good question. In order to put a name on a list, one has to know this name, i.e. recognize it. But I knew much more than 10 names. Actually, this is one of the topics I wanted to write about sometime in details. The whole atmosphere at that time was completely different from what I see around now. May be the place also played some role, but I doubt that its role was decisive. Most of the people around me liked to talk about mathematics, and not only about what they were doing. When some guy in Japan claimed that he proved the Riemann hypothesis, I knew about this the same week. Note that the internet was still in the future, as were e-mails. I had a feeling that I know about everything important going on in mathematics. I always had a little bit more curiosity than others, so I knew also about fields fairly remote from own work.
I do not remember all 10 names on my list (I remember about 7), but 4 winners were included. It was quite easy to guess 3 of them. Everybody would agree that they were the main contenders. I am really proud about guessing the 4th one. Nobody around was talking about him or even mentioned him, and his field is quite far from my own interests. To what extent I understood their work? I studied some work of one winner, knew the statements and had some idea about their proof for another one (later the work of both of them influenced a lot my own work, but mostly indirectly), and very well knew what are the achievements of the third one, why they are important, etc. I knew more or less just the statements of two main results of the 4th one, the one who was difficult to guess – for me. I was able to explain why this or that guy got the medal even to a theoretical physicist (actually did on one occasion). But I wasn’t able to teach a topic course about works of any of the 4.
At the time I never heard any complaints that a medal went to a wrong person. The same about all older awards. There was always a consensus in the mathematical community than all the people who got the medal deserved it. May be somebody else also deserved it too, but there are only 3 or 4 of them each time.
Mathematics is a human activity. This is one of the facts that T. Gowers prefers to ignore. Nobody verifies proofs line by line. Initially, you trust your guts feelings. If you need to use a theorem, you will be forced to study the proof and understand its main ideas. The same is true about the deepness of a result. You do not need to know all the proofs in order to write down a list like my list of 10 most likely winners (next time my list consisted of no more than 5 or 6, all winner were included). It seems that I knew the work of all guessed winners better than Gowers knew the work of 2010 medalists. But even if not, there is a huge difference between a graduate student trying to guess the current year winners, and a Fellow of the London Royal Society, a Fields medalist himself, who is deciding who will get 2010 medals. He should know more.
The job is surely not an easy one now, when it is all about politics. Otherwise it would be very pleasant.
Next post: Guessing who will get Fields medals - Some history and 2014.
I was asked by Tamas Gabal about possible 2014 Fields medalists listed in an online poll. I am neither ready to systematically write down my thoughts about the prizes in general and Fields medals in particular, nor to predict who will get 2014 medals. I am sure that the world would be better without any prizes, especially without Fields medals. Also, in my opinion, no more than two persons deserve 2014 Fields medals. Instead of trying to argue these points, I will quote my reply to Tamas Gabal (slightly edited).
Would I know who the members of the Fields medal committee are, I would be able to predict medalists with 99% confidence. But the composition of the committee is a secret. In the past, the situation was rather different. The composition of the committee wasn't important. When I was just a second year graduate student, I compiled a list of 10 candidates, among whom I considered 5 to have significantly higher chances (I never wrote down this partition, and the original list is lost for all practical purposes). All 4 winners were on the list. I was especially proud of predicting one of them; he was a fairly nontraditional at the time (or so I thought). I cannot do anything like this now without knowing the composition of the committee. Recent choices appear to be more or less random, with some obvious exceptions (like Grisha Perelman).
Somewhat later I wrote:
In the meantime I looked at the current results of that poll. Look like the preferences of the public are determined by the same mechanism as the preferences for movie actors and actresses: the name recognition.
Tamas Gabal replied:
Sowa, when you were a graduate student and made that list of possible winners, did you not rely on name recognition at least partially? Were you familiar with their work? That would be pretty impressive for a graduate student, since T. Gowers basically admitted that he was not really familiar with the work of Fields medalists in 2010, while he was a member of the committee. I wonder if anyone can honestly compare the depth of the work of all these candidates? The committee will seek an opinion of senior people in each area (again, based on name recognition, positions, etc.) and will be influenced by whoever makes the best case... It's not an easy job for sure.
Here is my reply.
Good question. In order to put a name on a list, one has to know this name, i.e. recognize it. But I knew much more than 10 names. Actually, this is one of the topics I wanted to write about sometime in details. The whole atmosphere at that time was completely different from what I see around now. May be the place also played some role, but I doubt that its role was decisive. Most of the people around me liked to talk about mathematics, and not only about what they were doing. When some guy in Japan claimed that he proved the Riemann hypothesis, I knew about this the same week. Note that the internet was still in the future, as were e-mails. I had a feeling that I know about everything important going on in mathematics. I always had a little bit more curiosity than others, so I knew also about fields fairly remote from own work.
I do not remember all 10 names on my list (I remember about 7), but 4 winners were included. It was quite easy to guess 3 of them. Everybody would agree that they were the main contenders. I am really proud about guessing the 4th one. Nobody around was talking about him or even mentioned him, and his field is quite far from my own interests. To what extent I understood their work? I studied some work of one winner, knew the statements and had some idea about their proof for another one (later the work of both of them influenced a lot my own work, but mostly indirectly), and very well knew what are the achievements of the third one, why they are important, etc. I knew more or less just the statements of two main results of the 4th one, the one who was difficult to guess – for me. I was able to explain why this or that guy got the medal even to a theoretical physicist (actually did on one occasion). But I wasn’t able to teach a topic course about works of any of the 4.
At the time I never heard any complaints that a medal went to a wrong person. The same about all older awards. There was always a consensus in the mathematical community than all the people who got the medal deserved it. May be somebody else also deserved it too, but there are only 3 or 4 of them each time.
Mathematics is a human activity. This is one of the facts that T. Gowers prefers to ignore. Nobody verifies proofs line by line. Initially, you trust your guts feelings. If you need to use a theorem, you will be forced to study the proof and understand its main ideas. The same is true about the deepness of a result. You do not need to know all the proofs in order to write down a list like my list of 10 most likely winners (next time my list consisted of no more than 5 or 6, all winner were included). It seems that I knew the work of all guessed winners better than Gowers knew the work of 2010 medalists. But even if not, there is a huge difference between a graduate student trying to guess the current year winners, and a Fellow of the London Royal Society, a Fields medalist himself, who is deciding who will get 2010 medals. He should know more.
The job is surely not an easy one now, when it is all about politics. Otherwise it would be very pleasant.
Next post: Guessing who will get Fields medals - Some history and 2014.
Wednesday, April 3, 2013
What is mathematics?
Previous post: D. Zeilberger's Opinions 1 and 62.
This is a reply to a comment by vznvzn to a post in this blog.
Next post: To appear.
This is a reply to a comment by vznvzn to a post in this blog.
I am not in the business of predicting the future. I have no idea what people will take seriously in 2050. I do not expect that Gowers’s fantasies, or yours, which are nearly the same, will turn into reality. I wouldn’t be surprised that the humanity will return by this time to the Middle Ages, or even to the pre-historical level. But I wouldn't bet on this even a dollar. At the same time mathematics can disappear very quickly. Mathematics is an extremely unusual and fragile human activity, which appeared by an accident, then disappeared for almost a thousand years, then slowly returned. The flourishing of mathematics in 20th century is very exceptional. In fact, we already have much more mathematicians (this is true independently of which meaning of the word “mathematician” one uses) than society, or, if you prefer, humanity needs.
The meaning of words “mathematics”, “mathematician” becomes important the moment the “computer-assisted proofs” are mentioned. Even Gowers agrees that if his projects succeeds, there will be no (pure) mathematicians in the current (or 300, or 2000 years old) sense. The issue can be the topic of a long discussion, of a serious monograph which will be happily published by Princeton University Press, but I am not sure that you are interested in going into it deeply. Let me say only point out that mathematics has any value only as human activity. It is partially a science, but to a big extent it is an art. All proofs belong to the art part. They are not needed at all for applications of mathematics. If a proof cannot be understood by humans (like the purported proofs in your examples), they have no value. Or, rather, their value is negative: a lot of human time and computer resources were wasted.
Now, a few words about your examples. The Kepler conjecture is not an interesting mathematical problem. It is not related to anything else, and its solution is isolated also. Physicists had some limited interest in it, but for them it obvious for a long time (probably, from the very beginning) that the conjectured result is true.
4 colors problem is not interesting either. Think for a moment, who may care if every map can be colored by only 4 colors? In the real word we have much more colors at our disposal, and in mathematics we have a beautiful, elementary, but conceptual proof of a theorem to the effect that 5 colors are sufficient. This proof deserves to be studied by every student of mathematics, but nobody rushed to study the Appel-Haken “proof” of 4-colors “theorem”. When a graduate student was assigned the task to study it (and, if I remember correctly, to reproduce the computer part for the first time), he very soon found several gaps. Then Haken published an amusing article, which can be summarized as follows. The “proof” has such a nature that it may have only few gaps and to find even one is extremely difficult. Therefore, if somebody did found a gap, it does not matter. This is so ridiculous that I am sure that my summary is not complete. Today it is much easier than at that time to reproduce the computer part, and the human part was also simplified (it consists in verifying by hand some properties of a bunch of graphs, more than 1,000 or even 1,500 in the Appel-Haken “proof”, less than 600 now.)
Wiles deserved a Fields medal not because he proved LFT; he deserved it already in 1990, before he completed his proof. In any case, the main and really important result of his work is not the proof of the LFT (this is for popular books), but the proof of the so-called modularity conjecture for nearly all cases (his students later completed the proof of the modularity conjecture for the exceptional cases). Due to some previous work by other mathematicians, all very abstract and conceptual, this immediately implies the LFT. Before this work (mid-1980ies), there was no reason even to expect that the LFT is true. Wiles himself learned about the LFT during his school years (every future mathematician does) and dreamed about proving it (only few have such dreams). But he did not move a finger before it was reduced to the modularity conjecture. Gauss, who was considered as King of Mathematics already during his lifetime, was primarily a number theorist. When asked, he dismissed the LFT as a completely uninteresting problem: “every idiot can invent zillions of such problems, simply stated, but requiring hundreds years of work of wise men to be solved”. Some banker already modified the LFT into a more general statement not following from the Wiles work and even announced a monetary prize for the proof of his conjecture. I am not sure, but, probably, he wanted a solution within a specified time frame; perhaps, there is no money for this anymore.
Let me tell you about another, mostly forgotten by now , example. It is relevant here because, like the 3x+1 problem (the Collatz conjecture), it deals with iterations of a simple rule and by another reason, which I will mention later. In other words, both at least formally belong to the field of dynamical systems, being questions about iterations.
My example is the Feigenbaum conjecture about iterations of maps of an interval to itself. Mitchell Feigenbaum is theoretical physicist, who was lead to his conjecture by physical intuition and extensive computer experiments. (By the way, I have no objections when computers are used to collect evidence.) The moment it was published, it was considered to be a very deep insight even as a conjecture and as a very important and challenging conjecture. The Feigenbaum conjecture was proved with some substantial help of computers only few years later by an (outstanding independently of this) mathematical physicists O. Lanford. For his computer part to be applicable, he imposed additional restrictions on the maps considered. Still, the generality is dear to mathematicians, but not to physicists, and the problem was considered to be solved. In a sense, it was solved indeed. Then another mathematician, D. Sullivan, who recently moved from topology to dynamical systems, took the challenge and proved the Feigenbaum conjecture without any assistance from the computers. This is quite remarkable by itself, mathematicians usually abandon problem or even the whole its area after a computer-assisted argument. Even more remarkable is the fact that his proof is not only human-readable, but provides a better result. He lifted the artificial Lanford’s restrictions.
The next point (the one I promised above) is concerned with standards Lanford applied to the computer-assisted proofs. He said and wrote that a computer-assisted proof is a proof in any sense only if the author not only understands its human-readable part, but also understands every line of the computer code (and can explain why this code does that is claimed it does). Moreover, the author should understand all details of the operations system used, up to the algorithm used to divide the time between different programs. For Lanford, a computer-assisted proof should be understandable to humans in every detail, except it may take too much time to reproduce the computations themselves.
Obviously, Lanford understood quite well that mathematics is a human activity.
Compare this with what is going on now. People freely use Windows OS (it seems that even at Microsoft nobody understands how and what it does), and the proprietary software like Mathematica™, for which the code is a trade secret and the reverse engineering is illegal. From my point of view, this fact alone puts everything done using this software outside not only of mathematics, but of any science.
Next post: To appear.
Tuesday, January 1, 2013
Reply to a comment
Previous post: Freedom of speech in mathematics
This is a reply to a recent comment by an Anonymous.
Dear Anonymous,
Thank you very much for reading these notes and for writing a serious and stimulating comment. I was thinking about it for a few days, and I am still not sure that my reply will be convincing. But let me try.
The initial goal of this blog was to provide a context for my arguments in Gowers's blog and in another blog. The latter was quoted by another anonymous in Gowers’s blog, and I had no other choice as to try to explain them (the original comment was addressed to people who know me and my views fairly well). So, the concentration on Gowers's views and activities was only natural; this was the intention. Now this topic is more or less exhausted, and expanding the scope of the blog, or even changing it completely may be in order.
Gowers himself described his goals and said that achieving them will eliminate mathematics as we know it. There is nothing apocalyptic in my discussion of his ideas. I do not anticipate that he or his followers will achieve his goals. I do not think that humans are some sort of computers, and I do not think that computers can do real mathematics (definitely, they can do a lot of mathematical things better than humans, but only when a class of problems was completely understood by humans). But he may achieve his goal in an Orwellian way by changing the meaning of the word “mathematics”. He already shifted the preferences of a big part of mathematical community. It took about ten years. If he will be able to do a comparable feat in the next ten years, and then, may be with his followers, once more, “mathematics” will mean “Hungarian-style combinatorics”. And I do believe that the Hungarian-style combinatorics is a field where computers eventually will be superior to humans because a nearly exhaustive search for a proof will be more efficient than human insights (like it happened with chess, which is actually a branch of combinatorics).
Independently of this scenario, I believe that having a person intended to eliminate mathematics (even if his goal is not achievable) in position of such influence as Gowers is extremely unhealthy and dangerous.
I agree that an “open and critical analysis of very influential individuals or groups in the mathematics research community” is highly desirable. But I am not quite comfortable with the way you wrote about this: “what the discussion should be about”. First of all, I am not comfortable with writing this either, but this is my blog and I write about whatever I like and whenever I like. Perhaps, you meant to apply “should” not to me, but to the mathematical community itself. I will assume that this is what you meant.
I believe that such a discussion is hardly possible. As a rule, mathematicians strongly dislike to openly discuss any contentious issues. One may see this everywhere: from insignificant issues on a department of mathematics to major decision made by the AMS or Fields Medals Committee, for example. I would suggest these posts for publication in the Notices of the AMS (under my real name) if I would believe that there is some chance for them to be published.
To Editors of the Notices of the AMS: this is a challenge. Prove that I am wrong!
Anyhow, I am willing to participate in such an open discussion. There is no venue for this now. Still, I would be glad to expand this blog into such a venue. For example, it may be easily transformed into a collective blog, and, for example, you will be able to post here. You will need only a Google account for this, and I will need to know the name of the account and some way of verification that it belongs to the Anonymous who wrote the comment. I reserve the right to be the owner of the blog and its moderator (right now it is not moderated, comments appear without my approval). And, may be, I will eventually transfer it to somebody else.
Finally, I disagree that mathematics was “long dominated by geometry, topology, arithmetic geometry, etc.” By some reason the word “geometry” is very popular for a few decades, and topology is usually considered to be a branch of geometry. So, the word “geometry” was appended to almost any good mathematical theory. Say, Non-Commutative Geometry is actually a branch of functional analysis. Arithmetic Geometry is a branch of algebraic number theory. Topology is not a branch of geometry in the classical sense of the word “geometry”. Algebraic Topology is a branch of algebra. Well, I realize that this is a sort of scandalous statement. It took me many years to come to this conclusion. There was a subfield of topology called Geometric Topology (it is hardly alive by now); this would be a nonsense would topology be indeed a branch of geometry.
So, in my opinion mathematics was long dominated by good mathematics, but this is not the case anymore.
Next post: Happy New Year!
This is a reply to a recent comment by an Anonymous.
Dear Anonymous,
Thank you very much for reading these notes and for writing a serious and stimulating comment. I was thinking about it for a few days, and I am still not sure that my reply will be convincing. But let me try.
The initial goal of this blog was to provide a context for my arguments in Gowers's blog and in another blog. The latter was quoted by another anonymous in Gowers’s blog, and I had no other choice as to try to explain them (the original comment was addressed to people who know me and my views fairly well). So, the concentration on Gowers's views and activities was only natural; this was the intention. Now this topic is more or less exhausted, and expanding the scope of the blog, or even changing it completely may be in order.
Gowers himself described his goals and said that achieving them will eliminate mathematics as we know it. There is nothing apocalyptic in my discussion of his ideas. I do not anticipate that he or his followers will achieve his goals. I do not think that humans are some sort of computers, and I do not think that computers can do real mathematics (definitely, they can do a lot of mathematical things better than humans, but only when a class of problems was completely understood by humans). But he may achieve his goal in an Orwellian way by changing the meaning of the word “mathematics”. He already shifted the preferences of a big part of mathematical community. It took about ten years. If he will be able to do a comparable feat in the next ten years, and then, may be with his followers, once more, “mathematics” will mean “Hungarian-style combinatorics”. And I do believe that the Hungarian-style combinatorics is a field where computers eventually will be superior to humans because a nearly exhaustive search for a proof will be more efficient than human insights (like it happened with chess, which is actually a branch of combinatorics).
Independently of this scenario, I believe that having a person intended to eliminate mathematics (even if his goal is not achievable) in position of such influence as Gowers is extremely unhealthy and dangerous.
I agree that an “open and critical analysis of very influential individuals or groups in the mathematics research community” is highly desirable. But I am not quite comfortable with the way you wrote about this: “what the discussion should be about”. First of all, I am not comfortable with writing this either, but this is my blog and I write about whatever I like and whenever I like. Perhaps, you meant to apply “should” not to me, but to the mathematical community itself. I will assume that this is what you meant.
I believe that such a discussion is hardly possible. As a rule, mathematicians strongly dislike to openly discuss any contentious issues. One may see this everywhere: from insignificant issues on a department of mathematics to major decision made by the AMS or Fields Medals Committee, for example. I would suggest these posts for publication in the Notices of the AMS (under my real name) if I would believe that there is some chance for them to be published.
To Editors of the Notices of the AMS: this is a challenge. Prove that I am wrong!
Anyhow, I am willing to participate in such an open discussion. There is no venue for this now. Still, I would be glad to expand this blog into such a venue. For example, it may be easily transformed into a collective blog, and, for example, you will be able to post here. You will need only a Google account for this, and I will need to know the name of the account and some way of verification that it belongs to the Anonymous who wrote the comment. I reserve the right to be the owner of the blog and its moderator (right now it is not moderated, comments appear without my approval). And, may be, I will eventually transfer it to somebody else.
Finally, I disagree that mathematics was “long dominated by geometry, topology, arithmetic geometry, etc.” By some reason the word “geometry” is very popular for a few decades, and topology is usually considered to be a branch of geometry. So, the word “geometry” was appended to almost any good mathematical theory. Say, Non-Commutative Geometry is actually a branch of functional analysis. Arithmetic Geometry is a branch of algebraic number theory. Topology is not a branch of geometry in the classical sense of the word “geometry”. Algebraic Topology is a branch of algebra. Well, I realize that this is a sort of scandalous statement. It took me many years to come to this conclusion. There was a subfield of topology called Geometric Topology (it is hardly alive by now); this would be a nonsense would topology be indeed a branch of geometry.
So, in my opinion mathematics was long dominated by good mathematics, but this is not the case anymore.
Next post: Happy New Year!
Sunday, May 20, 2012
The Politics of Timothy Gowers. 1
Previous post: My affair with Szemerédi-Gowers mathematics.
I mentioned in a comment in a blog that a substantial part of activity of Timothy Gowers in recent ten or more years is politics. It seems that this claim needs to be clarified. I will start with the definitions of the word “politics” in Merriam-Webster online. There are several meanings, of which the following (3a, 5a, 5b) are the most relevant.
3
a : political affairs or business; especially : competition between competing interest groups or individuals for power and leadership (as in a government).
5
a : the total complex of relations between people living in society
b : relations or conduct in a particular area of experience especially as seen or dealt with from a political point of view
It seems that the interpretation of T. Gowers himself is based only on the most objectionable meaning, namely:
3
c : political activities characterized by artful and often dishonest practices.
I am not in the position to judge how artful the politics of Gowers is; its results suggest that it is highly artful. But I have no reason to suspect any dishonest practices.
With only one exception, I was (and I am) observing Gowers activities only online (this includes preprints and publications, of course). I easily admit that in this way I may get a distorted picture. But this online-visible part does exist, and this part is mostly politics of mathematics, not mathematics itself.
I do classify as politics things like “The Princeton Companion to Mathematics”, which do not look as such at the first sight. This particular book gives a fairly distorted and at some places an incorrect picture of mathematics, and this is why I consider it as politics – it is an attempt to influence both the wide mathematical public and the mathematicians in power.
I was shocked by Gowers reply to the anonym2’s comment to his post “ICM2010 — Villani laudatio” in his blog. The Gowers blog at the time of 2010 Congress clearly showed that he has almost no idea about the work of mathematicians awarded Fields medals that year. But Gowers was a member of the committee selecting the medalists. “How it could be?” asked anonym2. The reply was very short: “No comment”. This lack of a response (or should I say “this very telling response”?) and the following it explanations of T. Tao clearly showed that the work of the Fields medals committee is now a pure politics, contrary to Tao’s assertion of the opposite. If the members of the committee do not understand the work of laureates, they were not able to base their choices on the substance of the works considered, and only the politics is left. In fact, nowadays it is rather easy to guess which member of the committee was a sponsor for which medalist. This was not the case in the past, and the predictions of the mathematical community were very close to the outcome. I myself, being only a second year graduate student, not even suspecting that there is any politics involved, was able to compile a list of 10 potential Fields medalist for that year, and all four actual medalists were on the list. The question of anonym2 "How could the mathematical community be so wrong in their predictions?" could not even arise at these times.
Next post: Part 2.
I mentioned in a comment in a blog that a substantial part of activity of Timothy Gowers in recent ten or more years is politics. It seems that this claim needs to be clarified. I will start with the definitions of the word “politics” in Merriam-Webster online. There are several meanings, of which the following (3a, 5a, 5b) are the most relevant.
3
a : political affairs or business; especially : competition between competing interest groups or individuals for power and leadership (as in a government).
5
a : the total complex of relations between people living in society
b : relations or conduct in a particular area of experience especially as seen or dealt with from a political point of view
It seems that the interpretation of T. Gowers himself is based only on the most objectionable meaning, namely:
3
c : political activities characterized by artful and often dishonest practices.
I am not in the position to judge how artful the politics of Gowers is; its results suggest that it is highly artful. But I have no reason to suspect any dishonest practices.
With only one exception, I was (and I am) observing Gowers activities only online (this includes preprints and publications, of course). I easily admit that in this way I may get a distorted picture. But this online-visible part does exist, and this part is mostly politics of mathematics, not mathematics itself.
I do classify as politics things like “The Princeton Companion to Mathematics”, which do not look as such at the first sight. This particular book gives a fairly distorted and at some places an incorrect picture of mathematics, and this is why I consider it as politics – it is an attempt to influence both the wide mathematical public and the mathematicians in power.
I was shocked by Gowers reply to the anonym2’s comment to his post “ICM2010 — Villani laudatio” in his blog. The Gowers blog at the time of 2010 Congress clearly showed that he has almost no idea about the work of mathematicians awarded Fields medals that year. But Gowers was a member of the committee selecting the medalists. “How it could be?” asked anonym2. The reply was very short: “No comment”. This lack of a response (or should I say “this very telling response”?) and the following it explanations of T. Tao clearly showed that the work of the Fields medals committee is now a pure politics, contrary to Tao’s assertion of the opposite. If the members of the committee do not understand the work of laureates, they were not able to base their choices on the substance of the works considered, and only the politics is left. In fact, nowadays it is rather easy to guess which member of the committee was a sponsor for which medalist. This was not the case in the past, and the predictions of the mathematical community were very close to the outcome. I myself, being only a second year graduate student, not even suspecting that there is any politics involved, was able to compile a list of 10 potential Fields medalist for that year, and all four actual medalists were on the list. The question of anonym2 "How could the mathematical community be so wrong in their predictions?" could not even arise at these times.
Next post: Part 2.
My affair with Szemerédi-Gowers mathematics
Previous post: The times of André Weil and the times of Timothy Gowers. 3.
I learned about Szemerédi’s theorem in 1978 from the Séminaire Bourbaki talk by Jean-Paul Touvenot “La démonstration de Furstenberg du théorème de Szemerédi sur les progressions arithmétiques”. As it is clear already from the title, the talk was devoted to the work of Furstenberg and not to the work of Szemerédi.
The theorem itself looked amusing, being a generalization of a very well known theorem of van der Warden. The latter one was, probably, known to every former student of a mathematical school in USSR and was usually considered as a nice toy and a good way to show smart and mathematically inclined kids how tricky the use of the mathematical induction could be. Nobody considered it as a really important theorem or as a result comparable with the main work of van der Warden.
But the fact that such a statement can be proved by an application of the theory of dynamical systems was really surprising. It looks like Bourbaki devoted a talk to this subject exactly for the sake of this unusual at the time application and not for the sake of the theorem itself. According to a maxim attributed to Yu.I. Manin, proofs are more important than theorems, and definitions are more important than proofs. I wholeheartedly agree. In any case, the work of Szemerédi was not reported at the Séminaire Bourbaki. I also was impressed by this application of dynamical systems and later read several initial chapters of Furstenberg’s book. But when I told about this to a young very promising expert in my area of mathematics, I got very cold reception: “This is not interesting at all”. Even references to Bourbaki and to the dynamical systems did not help. Now I think that we were both right. The theorem was not interesting because it was (and, apparently, still is) useless for anything but to proving its variations, and it is not sufficiently charming by itself (I think that the weaker van der Warden’s theorem is more charming). The theorem is interesting because it can be proved by tools completely alien to its natural context.
Then I more or less forgot about it, with a short interruption when Furstenberg’s book appeared.
Many years later I learned about T. Gowers from a famous and very remarkable mathematician, whom I will simply call M, short for Mathematician. In 1995 he told me about work of Gowers on Banach spaces, stressing that a great work may be completely unnoticed by the mathematical community. According to M, Gowers solved all open problems about Banach spaces. I had some mixed feelings about this claim and M’s opinion. May be Gowers indeed solved all problems of the Banach spaces theory (it seems that he did not), but who cares? For outsiders the theory of Banach spaces is a dead theory deserving a chapter in Bourbaki’s treatise because its basic theorems (about 80 years old) are exceptionally useful. On the other hand, Gowers was a Congress speaker in 1994, and this means that his work did not went unnoticed. In 1998 Gowers was awarded one of the four Fields medals for that year, quite unexpectedly to every mathematician with whom I discussed 1998 awards (M is not among them). It was also surprising that in his talk on the occasion of the award Gowers spoke not about his work on Banach spaces, but about a new approach to Szemerédi’s theorem. The approach was, in fact, not quite new: it extended the ideas of an early paper by K.-F. Roth on this topic (the paper is a few years earlier than his proof of what is known now as the Tue-Siegel-Roth theorem).
I trusted enough to M’s opinion to conclude that, probably, all work by Gowers deserves attention. So, I paid some attention to his work about Szemerédi’s theorem, but his paper looked technically forbidding (especially given that my main interests always were more or less at the opposite pole of pure mathematics). Then Gowers published a brilliantly written essay “Two cultures in mathematics”. He argued that the mainstream mathematics, best represented by the work of Serre, Atiyah, Grothendieck and their followers (and may be even Witten, despite he is not really a mathematician) is no more than a half of mathematics, “the first culture”, as he called it. Usually it is called “the conceptual mathematics”, since the new concepts are much more important to it than solutions of particular problems (as was already mentioned, the definitions are more important than proofs and theorems). Gowers argued that there is an equally important “second culture”. Apparently, it is best represented by the so-called “Hungarian combinatorics” and the work of Erdös and his numerous collaborators. In this mathematics of “the second culture”, the problems are stressed, the elementary (not involving abstract concepts, but may be very difficult) proofs are preferred, and no rigid structures (like the structure of a simple Lie algebra) are visible. Moreover, Gowers argued that both cultures are similar in several important aspects, despite this is very far from being transparent. A crucial part of his essay is devoted to outlining these similarities. All this was written in an excellent language at the level of best classical fiction literature, and appeared to be very convincing.
I decided to at least attempt to learn something from this “second culture”. Very soon I have had some good opportunities. T. Gowers was giving a series of lectures about his work on Szemerédi’s theorem in a not very far university. I decided to drive there (a roundtrip for each lecture) and attend the lectures. The lectures turned out to be exceptionally good. Then, after I applied some minor pressure to one of my colleagues, he agreed to give a series of lectures about some tools used by Gowers in his work. His presentation was also exceptionally good. I also tried to read relevant chapters in some books. All this turned out to be even more attractive than I expected. I decided to teach a graduate course in combinatorics, and attempted to include some Gowers-style stuff. The latter wasn’t really successful; the subject matter is much more technically difficult (and I do not mean the work of Szemerédi and Gowers) than would be appropriate. Anyhow, over the years I devoted significant time and efforts to familiarize myself with this “second culture” mathematics. This was interrupted both by mathematical reasons (it is nearly impossible to completely switch areas in the western mathematical community), and by some completely external circumstances.
When later I looked anew both at the “second culture” mathematics and at the theory of the “Two cultures in mathematics”, I could not help but to admit that they both lost their appeal. There is no second culture. The fact is that some branches of mathematics are not mature enough to replace assembling long proofs out of many similar pieces by a conceptual framework, making them less elementary, but more clear. The results of the second culture still looked isolated from the mainstream mathematics. I realized that the elementary combinatorial methods of proofs, characteristic for the purported second culture, occur everywhere (including my own work in “the first culture”). I would not say that they are always inevitable, but very often it is simpler to verify some fact by a combinatorial argument than to find a conceptual framework trivializing it.
Perhaps, my opinion about the “second culture” reached its peak on the day (April 8, 2004) of posting to the arXiv of the Green-Tao paper about arithmetic progression of primes. Prime numbers are the central notion of mathematics, and every new result about them is interesting. But gradually it became clear that the Green-Tao paper has nothing to do with primes. Green and Tao proved a generalization of Szemerédi’s theorem. By some completely independent results about primes due to Goldston and Yildirim, the set of primes satisfies the assumptions of the Green-Tao theorem. The juxtaposition of these two independent results leads to a nicely looking theorem. But anything new about primes is contained in the Goldston-Yildirim part, and not in Green-Tao part. This was a big disappointment.
So, the affair ended without any drama, in contrast with the novel “The End of the Affair” by Graham Greene.
Next post: The politics of Timothy Gowers. 1.
I learned about Szemerédi’s theorem in 1978 from the Séminaire Bourbaki talk by Jean-Paul Touvenot “La démonstration de Furstenberg du théorème de Szemerédi sur les progressions arithmétiques”. As it is clear already from the title, the talk was devoted to the work of Furstenberg and not to the work of Szemerédi.
The theorem itself looked amusing, being a generalization of a very well known theorem of van der Warden. The latter one was, probably, known to every former student of a mathematical school in USSR and was usually considered as a nice toy and a good way to show smart and mathematically inclined kids how tricky the use of the mathematical induction could be. Nobody considered it as a really important theorem or as a result comparable with the main work of van der Warden.
But the fact that such a statement can be proved by an application of the theory of dynamical systems was really surprising. It looks like Bourbaki devoted a talk to this subject exactly for the sake of this unusual at the time application and not for the sake of the theorem itself. According to a maxim attributed to Yu.I. Manin, proofs are more important than theorems, and definitions are more important than proofs. I wholeheartedly agree. In any case, the work of Szemerédi was not reported at the Séminaire Bourbaki. I also was impressed by this application of dynamical systems and later read several initial chapters of Furstenberg’s book. But when I told about this to a young very promising expert in my area of mathematics, I got very cold reception: “This is not interesting at all”. Even references to Bourbaki and to the dynamical systems did not help. Now I think that we were both right. The theorem was not interesting because it was (and, apparently, still is) useless for anything but to proving its variations, and it is not sufficiently charming by itself (I think that the weaker van der Warden’s theorem is more charming). The theorem is interesting because it can be proved by tools completely alien to its natural context.
Then I more or less forgot about it, with a short interruption when Furstenberg’s book appeared.
Many years later I learned about T. Gowers from a famous and very remarkable mathematician, whom I will simply call M, short for Mathematician. In 1995 he told me about work of Gowers on Banach spaces, stressing that a great work may be completely unnoticed by the mathematical community. According to M, Gowers solved all open problems about Banach spaces. I had some mixed feelings about this claim and M’s opinion. May be Gowers indeed solved all problems of the Banach spaces theory (it seems that he did not), but who cares? For outsiders the theory of Banach spaces is a dead theory deserving a chapter in Bourbaki’s treatise because its basic theorems (about 80 years old) are exceptionally useful. On the other hand, Gowers was a Congress speaker in 1994, and this means that his work did not went unnoticed. In 1998 Gowers was awarded one of the four Fields medals for that year, quite unexpectedly to every mathematician with whom I discussed 1998 awards (M is not among them). It was also surprising that in his talk on the occasion of the award Gowers spoke not about his work on Banach spaces, but about a new approach to Szemerédi’s theorem. The approach was, in fact, not quite new: it extended the ideas of an early paper by K.-F. Roth on this topic (the paper is a few years earlier than his proof of what is known now as the Tue-Siegel-Roth theorem).
I trusted enough to M’s opinion to conclude that, probably, all work by Gowers deserves attention. So, I paid some attention to his work about Szemerédi’s theorem, but his paper looked technically forbidding (especially given that my main interests always were more or less at the opposite pole of pure mathematics). Then Gowers published a brilliantly written essay “Two cultures in mathematics”. He argued that the mainstream mathematics, best represented by the work of Serre, Atiyah, Grothendieck and their followers (and may be even Witten, despite he is not really a mathematician) is no more than a half of mathematics, “the first culture”, as he called it. Usually it is called “the conceptual mathematics”, since the new concepts are much more important to it than solutions of particular problems (as was already mentioned, the definitions are more important than proofs and theorems). Gowers argued that there is an equally important “second culture”. Apparently, it is best represented by the so-called “Hungarian combinatorics” and the work of Erdös and his numerous collaborators. In this mathematics of “the second culture”, the problems are stressed, the elementary (not involving abstract concepts, but may be very difficult) proofs are preferred, and no rigid structures (like the structure of a simple Lie algebra) are visible. Moreover, Gowers argued that both cultures are similar in several important aspects, despite this is very far from being transparent. A crucial part of his essay is devoted to outlining these similarities. All this was written in an excellent language at the level of best classical fiction literature, and appeared to be very convincing.
I decided to at least attempt to learn something from this “second culture”. Very soon I have had some good opportunities. T. Gowers was giving a series of lectures about his work on Szemerédi’s theorem in a not very far university. I decided to drive there (a roundtrip for each lecture) and attend the lectures. The lectures turned out to be exceptionally good. Then, after I applied some minor pressure to one of my colleagues, he agreed to give a series of lectures about some tools used by Gowers in his work. His presentation was also exceptionally good. I also tried to read relevant chapters in some books. All this turned out to be even more attractive than I expected. I decided to teach a graduate course in combinatorics, and attempted to include some Gowers-style stuff. The latter wasn’t really successful; the subject matter is much more technically difficult (and I do not mean the work of Szemerédi and Gowers) than would be appropriate. Anyhow, over the years I devoted significant time and efforts to familiarize myself with this “second culture” mathematics. This was interrupted both by mathematical reasons (it is nearly impossible to completely switch areas in the western mathematical community), and by some completely external circumstances.
When later I looked anew both at the “second culture” mathematics and at the theory of the “Two cultures in mathematics”, I could not help but to admit that they both lost their appeal. There is no second culture. The fact is that some branches of mathematics are not mature enough to replace assembling long proofs out of many similar pieces by a conceptual framework, making them less elementary, but more clear. The results of the second culture still looked isolated from the mainstream mathematics. I realized that the elementary combinatorial methods of proofs, characteristic for the purported second culture, occur everywhere (including my own work in “the first culture”). I would not say that they are always inevitable, but very often it is simpler to verify some fact by a combinatorial argument than to find a conceptual framework trivializing it.
Perhaps, my opinion about the “second culture” reached its peak on the day (April 8, 2004) of posting to the arXiv of the Green-Tao paper about arithmetic progression of primes. Prime numbers are the central notion of mathematics, and every new result about them is interesting. But gradually it became clear that the Green-Tao paper has nothing to do with primes. Green and Tao proved a generalization of Szemerédi’s theorem. By some completely independent results about primes due to Goldston and Yildirim, the set of primes satisfies the assumptions of the Green-Tao theorem. The juxtaposition of these two independent results leads to a nicely looking theorem. But anything new about primes is contained in the Goldston-Yildirim part, and not in Green-Tao part. This was a big disappointment.
So, the affair ended without any drama, in contrast with the novel “The End of the Affair” by Graham Greene.
Next post: The politics of Timothy Gowers. 1.
Saturday, April 14, 2012
The times of André Weil and the times of Timothy Gowers. 3
Previous post: The times of André Weil and the times of Timothy Gowers. 2.
Now we can hardly say that mathematics is a useless science in the sense of G.H. Hardy. It contributes to the exploitation in various ways. For example, the theory of stochastic differential equations, a highly sophisticated branch of mathematics, is essential for the financial manipulations leading to a redistribution of wealth from the middle class to the top 1% of the population. The encryption schemes, designed by mathematicians and implemented by software engineers, prevent access of the general public to all sorts of artistic and intellectual goods. This is a new phenomenon, a result of the development of the Internet.
There is no need to detail the enormous contribution of mathematics to the business of extermination; it is obvious now (this wasn’t known to the general public when A. Weil wrote his article).
Mathematicians are not as free now as they were at the times of André Weil. There are (almost?) no more non-mathematical jobs which will earn a decent livehood and will leave enough energy for mathematical research. This situation is aggravated by the fact that if someone is not employed by a sufficiently rich university, then he or she has no access to the current mathematical literature, which is mostly electronic now, and, if sold to individuals, then the prices are set to be prohibitive. The access to these electronic materials (which cost almost nothing to the publishers to produce) is protected by the above mentioned encryption tools. The industry of the scientific publishing does not have publishing as its main activity any more. Its main business now is the restricting access to scientific papers by a combination of encryption, software, and lobbying for favorable to this industry laws. The main goal pursued is the transfer of the taxpayers dollars to the pockets of its executives and shareholders (this topics deserves a separate detailed discussion).
There are no Nobel prizes in mathematics, but there are many others. The Norwegian Abel prize is specifically intended to be a “Nobel prize” in mathematics. Long before it was established (the first one was awarded in 2003), another prize, the Fields medal, achieved incredible prestige and influence in mathematics, despite the negligible monetary award associated with it. In contrast with the Nobel and Abel prizes, the Fields medal may be awarded only to “young” mathematicians. The meaning of the word “young” was initially not specified, but the mathematical establishment slowly arrived at a precise definition. Somebody is young for the purposes of awarding a Fields medal, if he did not achieved the age 41 in the year of the International Congress of Mathematicians, at which the medal is to be awarded. The Congresses are hold every 4 years (only World War II caused an interruption). So, the persons born in the year of a Congress have additional 4 year to work and to have their work recognized.
Even if this stupid rule would be discarded, the age limitation tends to reward fast people strong at applying existing methods to famous problems. The Fields medals (and many other prizes in mathematics) are usually awarded to the mathematician who made the last step in a solution of a problem, and only rarely to the one who discovered a new method or new line of thought. There are only little chances for “slow maturing work” to be rewarded by this most prestigious award (more prestigious by an order of magnitude than any other prize, except, may be, the Abel prize, which is up to now was awarded almost exclusively to the people of the retirement age).
It was possible to ignore all the prizes in 1948. The Fields medals were awarded only once, in 1936, to two mathematicians. Other prizes, where they existed, did not carry any serious prestige. But in 1950, 1954, and 1958 Fields medals went to exceptionally brilliant mathematicians, and since then this was a prize coveted by anybody who thought that there is a chance to get it.
Now there are many other prizes, each one striving to carry as much weight and influence as possible. An interesting example is the story of the Salem prize. The Salem prize was established by the widow of Raphaël Salem in order to encourage work in Salem's field of interest, primarily the theory of Fourier series. Note that Fourier series and their versions are used throughout almost whole mathematics; it is only natural to think that the prize was intended to mathematicians working on problems really close to Salem’s interests. The international committee (occasionally changing by an unknown to the public mechanism) gradually increased the scope of the prize. By 1991 no connection with Salem’s interests could be observed. Now it is the most prestigious prize for young analysts without any restrictions (and the analysis is understood in a very broad sense).
In fact, this change (as also a suspected preference for mathematicians belonging to one or two particular schools) was not welcomed by Salem’s family, and it withdraw the funding for the prize. The committee did not inform the mathematical public about these events and continued to award the prize with $0.00 attached. I am not aware of the current situation; may be the committee managed to raise some money. (Please, note that I cannot name my sources, as it is often the case in the news reporting, and hence cannot provide any proof. I can only vouch that my sources are reliable and well informed.)
The negligible monetary value of most mathematical prizes is not of any importance. The prestige is immediately transformed into the salary rises, offers from rich universities capable of doubling the salary, etc. The lifetime income could be increased by a much bigger amount than the monetary value of a Nobel Prize.
These are the signs of the lost innocence directly related to the article of André Weil. There are many other signs, and one can talk about them indefinitely. In any case, there are no more ivory towers for mathematicians; their jobs depend on many complicated and not always natural implicit agreements in the society, various laws and regulations detailing the laws, etc. From 1945 till about 1985 all these agreements and laws worked very favorably for mathematics. But, as it turned out, the same laws and understandings could be easily used to control mathematicians, sometimes directly, sometimes in hardly discernible ways, and the same arguments that were used to increase the number of jobs 60 or 50 years ago, could, in principle, be used to eliminate these jobs completely.
Next post: My affair with Szemerédi-Gowers mathematics.
Now we can hardly say that mathematics is a useless science in the sense of G.H. Hardy. It contributes to the exploitation in various ways. For example, the theory of stochastic differential equations, a highly sophisticated branch of mathematics, is essential for the financial manipulations leading to a redistribution of wealth from the middle class to the top 1% of the population. The encryption schemes, designed by mathematicians and implemented by software engineers, prevent access of the general public to all sorts of artistic and intellectual goods. This is a new phenomenon, a result of the development of the Internet.
There is no need to detail the enormous contribution of mathematics to the business of extermination; it is obvious now (this wasn’t known to the general public when A. Weil wrote his article).
Mathematicians are not as free now as they were at the times of André Weil. There are (almost?) no more non-mathematical jobs which will earn a decent livehood and will leave enough energy for mathematical research. This situation is aggravated by the fact that if someone is not employed by a sufficiently rich university, then he or she has no access to the current mathematical literature, which is mostly electronic now, and, if sold to individuals, then the prices are set to be prohibitive. The access to these electronic materials (which cost almost nothing to the publishers to produce) is protected by the above mentioned encryption tools. The industry of the scientific publishing does not have publishing as its main activity any more. Its main business now is the restricting access to scientific papers by a combination of encryption, software, and lobbying for favorable to this industry laws. The main goal pursued is the transfer of the taxpayers dollars to the pockets of its executives and shareholders (this topics deserves a separate detailed discussion).
There are no Nobel prizes in mathematics, but there are many others. The Norwegian Abel prize is specifically intended to be a “Nobel prize” in mathematics. Long before it was established (the first one was awarded in 2003), another prize, the Fields medal, achieved incredible prestige and influence in mathematics, despite the negligible monetary award associated with it. In contrast with the Nobel and Abel prizes, the Fields medal may be awarded only to “young” mathematicians. The meaning of the word “young” was initially not specified, but the mathematical establishment slowly arrived at a precise definition. Somebody is young for the purposes of awarding a Fields medal, if he did not achieved the age 41 in the year of the International Congress of Mathematicians, at which the medal is to be awarded. The Congresses are hold every 4 years (only World War II caused an interruption). So, the persons born in the year of a Congress have additional 4 year to work and to have their work recognized.
Even if this stupid rule would be discarded, the age limitation tends to reward fast people strong at applying existing methods to famous problems. The Fields medals (and many other prizes in mathematics) are usually awarded to the mathematician who made the last step in a solution of a problem, and only rarely to the one who discovered a new method or new line of thought. There are only little chances for “slow maturing work” to be rewarded by this most prestigious award (more prestigious by an order of magnitude than any other prize, except, may be, the Abel prize, which is up to now was awarded almost exclusively to the people of the retirement age).
It was possible to ignore all the prizes in 1948. The Fields medals were awarded only once, in 1936, to two mathematicians. Other prizes, where they existed, did not carry any serious prestige. But in 1950, 1954, and 1958 Fields medals went to exceptionally brilliant mathematicians, and since then this was a prize coveted by anybody who thought that there is a chance to get it.
Now there are many other prizes, each one striving to carry as much weight and influence as possible. An interesting example is the story of the Salem prize. The Salem prize was established by the widow of Raphaël Salem in order to encourage work in Salem's field of interest, primarily the theory of Fourier series. Note that Fourier series and their versions are used throughout almost whole mathematics; it is only natural to think that the prize was intended to mathematicians working on problems really close to Salem’s interests. The international committee (occasionally changing by an unknown to the public mechanism) gradually increased the scope of the prize. By 1991 no connection with Salem’s interests could be observed. Now it is the most prestigious prize for young analysts without any restrictions (and the analysis is understood in a very broad sense).
In fact, this change (as also a suspected preference for mathematicians belonging to one or two particular schools) was not welcomed by Salem’s family, and it withdraw the funding for the prize. The committee did not inform the mathematical public about these events and continued to award the prize with $0.00 attached. I am not aware of the current situation; may be the committee managed to raise some money. (Please, note that I cannot name my sources, as it is often the case in the news reporting, and hence cannot provide any proof. I can only vouch that my sources are reliable and well informed.)
The negligible monetary value of most mathematical prizes is not of any importance. The prestige is immediately transformed into the salary rises, offers from rich universities capable of doubling the salary, etc. The lifetime income could be increased by a much bigger amount than the monetary value of a Nobel Prize.
These are the signs of the lost innocence directly related to the article of André Weil. There are many other signs, and one can talk about them indefinitely. In any case, there are no more ivory towers for mathematicians; their jobs depend on many complicated and not always natural implicit agreements in the society, various laws and regulations detailing the laws, etc. From 1945 till about 1985 all these agreements and laws worked very favorably for mathematics. But, as it turned out, the same laws and understandings could be easily used to control mathematicians, sometimes directly, sometimes in hardly discernible ways, and the same arguments that were used to increase the number of jobs 60 or 50 years ago, could, in principle, be used to eliminate these jobs completely.
Next post: My affair with Szemerédi-Gowers mathematics.
Subscribe to:
Posts (Atom)