What is combinatorics and what this blog is about according to Igor Pak

Previous post: About Timothy Gowers.

I came across the post “What is Combinatorics?” by Igor Pak. His intention seems to be refuting what is, in his opinion, a basic fault of my notes, namely, the lack of understanding of what is combinatorics.

“While myself uninterested in engaging in conversation, I figured that there got to be some old “war-time” replies which I can show to the Owl blogger.  As I see it, only the lack of knowledge can explain these nearsighted generalizations the blogger is showing.  And in the age of Google Scholar, there really is no excuse for not knowing the history of the subject, and its traditional sensitivities.”

Unfortunately, he did not show me anything. I come across his post while searching other things by Google. May be he is afraid that giving me a link in a comment will engage him in conversation? I would be glad to discuss these issues with him, but if he is not inclined, how can I insist? My intention was to write a comment in his blog, but for this one needs to be registered at WordPress.com. Google is more generous, as is T. Gowers, who allows non-WordPress comments in his blog.

Indeed, I don't know much about “traditional sensitivities” of combinatorics. A Google search resulted in links to his post and to numerous papers about “noise sensitivity”.

Beyond this, he is fighting windmills. I agree with most of what he wrote. Gian-Carlo Rota is my hero also. But I devoted a lot of time and space to explaining what I mean by "combinatorial" mathematics, and even stated that I use this term only because it is used by Gowers (and all my writings on this topics have a root in his ones), and I wasn't able to find quickly a good replacement (any suggestions?). See, for example, the beginning of the post “The conceptual mathematics vs. the classical (combinatorial) one” , as also other posts and my comments in Gowers's blog. In particular, I said that there is no real division between Gowers's “second culture” and “first culture”, and therefore there is no real division between combbinatorics and non-combinatorics.

So, for this blog the working definition of combinatorics is “branches of mathematics described in two essays by T. Gowers as belonging to the second culture and opposed in spirit to the Grothendieck's mathematics”.

I don't like much boxing of all theorems or papers into various classes, be they invented by AMS, NSF, or other “authorities”. I cannot say what is my branch of mathematics. Administrators usually assign to me the field my Ph.D. thesis belongs to, but I did not worked in it since then. I believe that the usual division of mathematics into Analysis, Algebra, Combinatorics, Geometry, etc. is hopelessly outdated.


Next post: New comments to the post "What is mathematics?"

The Hungarian Combinatorics from an Advanced Standpoint

Previous post: Conceptual mathematics vs. the classical (combinatorial) one.

Again,  this post is a long reply to questions posed by ACM. It is a complement to the previous post "Conceptual mathematics vs. the classical (combinatorial) one". The title is intentionally similar to the titles of three well known books by F. Klein.

First, the terminology in “Conceptual mathematics vs. the classical (combinatorial) one” is my and was invented at the spot, and the word "classical" is a very bad choice. I should find something better. The word "conceptual" is good enough, but not as catchy as I may like. I meant something real, but as close as possible to the Gowers's idea of "two cultures". I do not believe in his theory anymore; but by simply using his terms I will promote it.

Another choice, regularly used in discussions in Gowers's blog is "combinatorial". It looks like it immediately leads to confusion, as one may see from your question (but not only). First of all (I already mentioned it in Gowers's blog or here), there two rather different types of combinatorics. At one pole there is the algebraic combinatorics and most of the enumerative combinatorics. R. Stanley and the late J.-C. Rota are among the best (or the best) in this field. One can give even a more extreme example, mentioned by M. Emerton: symmetric group and its representations. Partitions of natural numbers are at the core of this theory, and in this sense it is combinatorics. One the other hand, it was always considered as a part of the theory of representations, a highly conceptual branch of mathematics.

So, there is already a lot of conceptual and quite interesting combinatorics. And the same time, there is Hungarian combinatorics, best represented by the Hungarian school. It is usually associated with P. Erdös and since the last year Abel prize is also firmly associated with E. Szemerédi. Currently T. Gowers is its primary spokesperson, with T. Tao serving as supposedly independent and objective supporter. Of course, all this goes back for centuries.

Today the most obvious difference between these two kinds of combinatorics is the fact that the algebraic combinatorics is mostly about exact values and identities, and Hungarian combinatorics is mostly about estimates and asymptotics. If no reasonable estimate is in sight, the existence is good enough. This is the case with the original version of Szemerédi's theorem. T. Gowers added to it some estimates, which are huge but a least could be written down by elementary means. He also proved that any estimate should be huge (in a precise sense). I think that the short paper proving the latter (probably, it was Gowers's first publication in the field) is the most important result around Szemerédi’s theorem. It is strange that it got almost no publicity, especially if compared with his other papers and Green-Tao's ones. It could be the case that this opinion results from the influence of a classmate, who used to stress that lower estimates are much more deep and important than the upper ones (for positive numbers, of course), especially in combinatorial problems.

Indeed, I do consider Hungarian combinatorics as the opposite of all new conceptual ideas discovered during the last 100 years. This, obviously, does not mean that the results of Hungarian combinatorics cannot be approached conceptually. We have an example at hand: Furstenberg’s proof of Szemerédi theorem. It seems that it was obtained within a year of the publication of Szemerédi’s theorem (did not checked right now). Of course, I cannot exclude the possibility that Furstenberg worked on this problem (or his framework for his proof without having this particular application as the main goal) for years within his usual conceptual framework, and missed by only few months. I wonder how mathematics would look now if Furstenberg would be the first to solve the problem.

One cannot approach the area (not the results alone) of Hungarian combinatorics from any conceptual point of view, since the Hungarian combinatorics is not conceptual almost by the definition (definitely by its description by Gowers in his “Two cultures”). I adhere to the motto “Proofs are more important than theorems, definitions are more important than proofs”. In fact, I was adhering to it long before I learned about this phrase; this was my taste already in the middle school (I should confess that I realized this only recently). Of course, I should apply it uniformly. In particular, the Hungarian style of proofs (very convoluted combinations of well known pieces, as a first approximation) is more essential than the results proved, and the insistence on being elementary but difficult should be taken very seriously – it excludes any deep definitions.

I am not aware of any case when “heuristic” of Hungarian combinatorics lead anybody to conceptual results. The theorems can (again, Furstenberg), but they are not heuristics.

I am not in the business of predicting the future, but I see only two ways for Hungarian combinatorics, assuming that the conceptual mathematics is not abandoned. Note that still not even ideas of Grothendieck are completely explored, and, according to his coauthor J. Dieudonne, there are enough ideas in Grothendieck’s work to occupy mathematicians for centuries to come – the conceptual mathematics has no internal reasons to die in any foreseeable future. Either the Hungarian combinatorics will mature by itself and will develop new concepts which eventually will turn it into a part of conceptual mathematics. There are at least germs of such development. For example, matroids (discovered by H. Whitney, one of the greatest topologists of the 20^th century) are only at the next level of abstraction after the graphs, but matroids is an immensely useful notion (unfortunately, it is hardly taught anywhere, which severely impedes its uses). Or it will remain a collection of elementary tricks, and will resemble more and more the collection of mathematical Olympiads problems. Then it will die out and forgotten.

I doubt that any area of mathematics, which failed to conceptualize in a reasonable time, survived as an active area of research. Note that the meaning of the word “reasonable” changes with time itself; at the very least because of the huge variations of the number of working mathematicians during the history. Any suggestions of counterexamples?

Next post: About Timothy Gowers.

The conceptual mathematics vs. the classical (combinatorial) one.

Previous post: Simons's video protection, youtube.com, etc.

This post is an attempt to answer some questions of ACM in a form not requiring knowledge of Grothendieck ideas or anything simlilar.

But it is self-contained and touches upon important and hardly wide known issues.

--------------------------------------------

It is not easy to explain how conceptual theorems and proofs, especially the ones of the level close to the one of Grothendieck work, could be at the same time more easy and more difficult at the same time. In fact, they are easy in one sense and difficult in another. The conceptual mathematics depends on – what one expect here? – on new concepts, or, what is the same, on the new definitions in order to solve new problems. The hard part is to discover appropriate definitions. After this proofs are very natural and straightforward up to being completely trivial in many situations. They are easy. Classically, the convoluted proofs with artificial tricks were valued most of all. Classically, it is desirable to have a most elementary proof possible, no matter how complicated it is.

A lot of efforts were devoted to attempts to prove the theorem about the distribution of primes elementary. In this case the requirement was not to use the theory of complex functions. Finally, such proof was found, and it turned out to be useless. Neither the first elementary proof, nor subsequent ones had clarified anything, and none helped to prove a much more precise form of this theorem, known as Riemann hypothesis (this is still an open problem which many consider as the most important problem in mathematics).

Let me try to do this using a simple example, which, perhaps, I had already mentioned (I am sure that I spoke about it quite recently, but it may be not online). This example is not a “model” or a toy, it is real.

Probably, you know about the so-called Fundamental Theorem of Calculus, usually wrongly attributed to Newton and Leibniz (it was known earlier, and, for example, was presented in the lectures and a textbook of Newton's teacher, John Barrow). It relates the derivatives with integrals. Nothing useful can be done without it. Now, one can integrate not only functions of real numbers, but also functions of two variables (having two real numbers as the input), three, and so on. One can also differentiate functions of several variables (basically, by considering them only along straight lines and using the usual derivatives). A function of, say, 5 variables has 5 derivatives, called its partial derivatives.

Now, the natural question to ask is if there is an analogue of the Fundamental Theorem of Calculus for functions of several variables. In 19^th century such analogues were needed for applications. Then 3 theorems of this sort were proved, namely, the theorems of Gauss-Ostrogradsky (they discovered it independently of each other, and I am not sure if there was a third such mathematician or not), Green, and Stokes (some people, as far as I remember, attribute it to J.C. Maxwell, but it is called the Stokes theorem anyhow). The Gauss-Ostrogradsky theorem deals with integration over 3-dimensional domains in space, the Green theorem with 2 dimensional planar domains, and the Stokes theorem deals with integration over curved surfaces in the usual 3-dimensional space. I hope that I did not mix them up; the reason why this could happen is at the heart of the matter. Of course, I can check this in moment; but then an important point would be less transparent.

Here are 3 theorems, clearly dealing with similar phenomena, but looking very differently and having different and not quite obvious proofs. But there are useful functions of more than 3 variables. What about them? There is a gap in my knowledge of the history of mathematics: I don’t know any named theorem dealing with more variables, except the final one. Apparently, nobody wrote even a moderately detailed history of the intermediate period between the 3 theorems above and the final version.

The final version is called the Stokes theorem again, despite Stokes has nothing do with it (except that he proved that special case). It applies to functions of any number of variables and even to functions defined on so-called smooth manifolds, the higher-dimensional generalization of surfaces. On manifolds, variables can be introduced only locally, near any point; and manifolds themselves are not assumed to be contained in some nice ambient space like the Euclidean space. So, the final version is much more general. And the final version has exactly the same form in all dimension, but the above mentioned 3 theorems are its immediate corollaries. This is why it is so easy to forget which names are associated to which particular case.

And – surprise! – the proof of general Stokes theorem is trivial. There is a nice short (but very dense) book “Calculus on manifolds” by M. Spivak devoted to this theorem.  I recommend reading its preface to anybody interested in one way or another in mathematics. For mathematicians to know its content is a must. In the preface M. Spivak explains what happened. All the proofs are now trivial because all the difficulties were transferred into definitions. In fact, this Stokes theorem deals with integration not of functions, but of the so-called differential form, sometimes called also exterior forms. And this is a difficult notion. It requires very deep insights to discover it, and it still difficult to learn it. In the simplest situation, where nothing depends on any variables, it was discovered by H. Grassmann in the middle of 19^th century. The discoveries of this German school teacher are so important that the American Mathematical Society published an English translation of one of his books few years ago. It is still quite a mystery how he arrived at his definitions. With the benefits of hindsight, one may say that he was working on geometric problems, but was guided by the abstract algebra (which did not exist till 1930). Later on his ideas were generalized in order to allow everything to depend on some variables (probably, E. Cartan was the main contributor here). In 1930ies the general Stokes theorem was well known to experts. Nowadays, it is possible to teach it to bright undergraduates in any decent US university, but there are not enough of such bright undergraduates. It should be in some of the required course for graduate students, but one can get a Ph.D. without being ever exposed to it.

To sum up, the modern Stokes theorem requires learning a new and not very well motivated (apparently, even the Grassmann did not really understood why he introduced his exterior forms) notion of differential forms and their basic properties. Then you have a theorem from which all 19^th century results follow immediately, and which is infinitely more general than all of them together. At the same time it has the same form for any number of variables and has a trivial proof (and the proofs of the needed theorems about differential forms are also trivial). There are no tricks in the proofs; they are very natural and straightforward. All difficulties were moved into definitions.

Now, what is hard and what is difficult? New definitions of such importance are infinitely rarer than new theorems. Most mathematicians of even the highest caliber did not discover any such definition. Only a minority of Abel prize winner discovered anything comparable, and it is still too early to judge if their definitions are really important. So, discovering new concepts is hard and rare. Then there is a common prejudice against anything new (I am amazed that it took more than 15 years to convince public to buy HD TV sets, despite they are better in the most obvious sense), and there is a real difficulties in learning these new notions. For example, there is a notion of a derived category (it comes from the Grothendieck school), which most of mathematicians consider as difficult and hardly relevant. All proofs in this theory are utterly trivial.

Final note: the new conceptual proofs are often longer than the classical proofs even of the same results. This is because in the classical mathematics various tricks leading to shortcut through an argument are highly valued, and anything artificial is not valued at all in the conceptual mathematics.

Next post: The Hungarian Combinatorics from an Advanced Standpoint.

Combinatorics is not a new way of looking at mathematics

Previous post: The value of insights and the identity of the author.


This is a partial reply to a comment by vznvzn in Gowers's blog.


Combinatorics is most resolutely not "a new way of looking at mathematics". It is very old, definitely known for hundreds years. Perhaps, it was known in the ancient Babylon already.

And Erdős is not a "contrarian". His work belongs to the most widely practiced tradition in analysis. As a crude approximation, one can say that this tradition originates in the calculus of Leibniz, which is quite different from the calculus of Newton. Even most of mathematicians are not aware of the difference between the Leibniz calculus and the Newton calculus. This is not surprising at all, since only the Leibniz calculus is taught nowadays.

It is the Grothendieck's way of looking at mathematics, the one which I advocate, which is new. This new, conceptual, way of doing mathematics immediately met strong resistance.

And in some cases its opponents won. For example, the early work of Grothedieck in functional analysis had no influence till analysts managed to translate part of his ideas into their standard language. It seems that only quite recently some of analysts realized that a lot was lost in this translation, and done a better translation, closer to the spirit of the original work of Grothendieck.

Another example is provided by the invasion of this new style and even some technical concepts developed in this style into the analysis of several complex variables. This was intolerable for the classical complex analysts, and they started to stress problems about which it was more or less clear that they can be approached by familiar methods. They succeeded, and already in the 1970ies a prominent representative of the classical school, W. Rudin, was able proudly say that Grothendieck's methods (he was more specific) disappeared into background. He did not publish his opinion at the time, but attempted to insult a prominent representative of the new style, A. Borel by such statements. A quarter of century (or more) later he told this story in an autobiographical book. (W. Rudin is a good mathematician and the author of several exceptionally good books, but A. Borel was a brilliant mathematician.)

Now we are observing a much broader attempt, apparently led by T. Gowers, to eliminate the conceptual way of doing mathematics completely. At the very least T. Gowers is the face of this movement for the mathematical public.  After this T. Gowers envisions an elimination of the mathematics itself by relegating it to computers. It looks like the second step is the one most dear to his heart (see the discussion in his blog about a year ago). It seems that combinatorics is much more amenable to the computerization (although I don't believe that even this is possible) than the conceptual mathematics.

Actually, it is not hard to believe that computers can efficiently produce proofs of a wide class of theorem (the proofs will be unreadable to humans, but still some will consider them as proofs). But for the conceptual mathematics it is the definition, and not the proofs, which is important. The conceptual mathematics is looking for new definitions interesting to humans. The proof and theorems serve as a stimulus for work and as a necessary testing ground for new definitions. If a new definition does not help to prove new theorems or to simplify the proofs of old ones, it is not interesting for humans.

There is only one way to get rid of the conceptual mathematics, namely, the Wigner shift of the second kind. The new generation should be told that combinatorics is new, that it is the field to work in, and very soon we will see the young people only the ones doing combinatorics. Since mathematics is to a huge extent "a young people’s game", such a shift can be accomplished very quickly.

P.S. It is worth to note that there are two branches of combinatorics, and one of them is already belongs to the conceptual mathematics. Some people (like D. Zeilberger) are intentionally ignoring this to promote the non-conceptual kind.



Next post: D. Zeilberger's Opinions 1 and 62.

T. Gowers about replacing mathematicians by computers. 2

Previous post: T. Gowers about replacing mathematicians by computers. 1.


As we do know too well by now, not all scientific or technological progress is unqualifiedly beneficial for the humanity. As one of the results of scientific research the humanity now has the ability to exterminate not only all humans, but also all the life on Earth. Dealing with this problem determined to a big extent the direction of development of western countries since shortly after WWII. There are not so dramatic examples also; a scientific research about humans may damage only minor part of the population, or even just the subjects of this research (during the last decades, such a research is carefully monitored in order to avoid any harm to the subjects).

Gowers’s project is an experiment on humans. I believe that replacing mathematicians by computers will do a lot of harm at least to the people who could find their joy and the meaning of life in doing mathematics. But the results, if the project succeeds, are not predictable. If we agree, together with André Weil, that mathematics is an indispensable part of our culture, then it hardly possible to predict what will happen without it.


There is also question if Gowers’s goal is achievable at all. He limited it in at least two significant respects. First, he would be satisfied even if computer will not surpass humans (as opposed to the designers of “Deep Blue”, who wanted and managed to surpass the best chess players). Second, he always speaks about proving theorems, and never about discovering analogies, introducing new definitions, etc. These aspects are the most important part of mathematics, not the theorems (compare the already quoted maxim by Manin). But only theorems matter in the Hungarian-style mathematics. Perhaps, this is the reason why Gowers never mentions these aspects of mathematics. It is hard to tell if this limited goal can be achieved. Given a statement, a computer definitely able sometimes to find a proof of it (or disprove it) by a sufficiently exhaustive search. If it is not able to give an answer, the problem remains open, exactly as in human mathematics. What kind of statements a computer will be able to deal with, is another question.

Some of the best problems are not a true-false type of questions. For example, the problem of defining a “good” cohomology theory for algebraic varieties over finite fields (to a big extent solved by Grothendieck), or the problem of defining higher algebraic K-functors (solved by Quillen). It is impossible for me to imagine a computer capable to invent new definitions or suggest problems based on vague analogies like these two problems, responsible for perhaps a half of really good mathematics after 1950.


It seems that I could feel safe: even in the gloomy Gowers’s future, there will be place for human mathematicians. In fact, the future theorems, stated as conjectures, always served as one of the main, or simply the main stimulus to invention of new definitions. In addition, the success of Gowers’s project will mean the end of mathematics as a profession. There will be no new mathematicians, of Serre’s level, or any other, simply because there will be no way to earn a living by doing human mathematics.

Next post: The twist ending. 1

T. Gowers about replacing mathematicians by computers. 1

Previous post: The Politics of Timothy Gowers. 3.


Starting with his “GAFA Visions” essay, T. Gowers promotes the idea that it is possible and desirable to design computers capable of proving theorems at a very high level, although he will be satisfied if such computers still will be not able to perform at the level of the very best mathematician, for example, at the level of Serre or Milnor. I attempted to discuss this topic with him in the comments to his post about this year Abel prize.

I had no plans for such a discussion, and the topic wasn’t selected by me. I made a spontaneous comment in another blog, which was a reaction to a reaction to a post about E. Szemerédi being awarded this year Abel prize. But I stated my position with many details in Gowers’s blog. T. Gowers replied to only three of my comments, and only partially. It seems that for many people it is hard to believe that a mathematician of the stature of T. Gowers may be interested in eliminating mathematics as a human activity, and this is why my comments in that blog made their way to Gowers’s one (one can find links in the latter).

For Gowers, the goal of designing computers capable of replacing mathematicians is fascinating by itself. Adding some details to his motivation, he claims that such computers cannot be designed without deep understanding of how humans prove theorems. He will not consider his goal achieved if the theorem-proving computer will operate in the manner of “Deep Blue” chess-playing computer, namely, by a huge and a massively parallel (like “Deep Blue”) search. Without any explanation, even after directly asked about this, he claims that in fact a computer operating in the manner of “Deep Blue” cannot be successful in proving theorems. In his opinion, such a computer should closely imitate humans (whence we will learn something about humans by designing such a computer), and that it is much simpler to imitate humans doing mathematics than other tasks.

In addition, Gowers holds the opinion that elimination of mathematics would be not a big loss, comparing it to losing many old professions to the technology.


Gowers’s position contradicts to the all the experience of the humanity. None of successful technologies imitates the way the humans act. No means of transportation imitates walking or running, for example. On the other end and closer to mathematics, no computer playing chess imitates human chess players.

Note that parallel processing (on which “Deep Blue” had heavily relied) is exactly that Gowers attempts to do with mathematics in his Polymath project. It seems that this project approaches the problem from the other end: it is an attempt to make humans to act like computers. This will definitely simplify the goal of imitating them by computers. Will they be humans after this?


Gowers’s position is a position of a scientist interested in learning how something functions and not caring about the cost; in his case not caring about the very survival of mathematics. In my opinion, this means that he is not a mathematician anymore. Of course, he proves theorems, relies on his mathematical experience in his destructive project, but these facts are uninteresting trivialities. I expect from mathematician affection toward mathematics and a desire of its continuing flourishing. (How many nominal mathematicians such a requirement will disqualify?)


Next post: T. Gowers about replacing mathematicians by computers. 2.

The Politics of Timothy Gowers. 2

Previous post: The Politics of Timothy Gowers. 1.


Since about 2000, T. Gowers became a prominent advocate of two ideas. First, he works on changing the mathematical public opinion about relative merits of various mathematical results and branches of mathematics in favor of his own area of expertise. Second, he advocates the elimination of mathematics as a significant human activity, and a gradual replacement of mathematicians by computers and moderately skilled professionals assisting these computers. The second goal is more remote in time; he estimates that it is at least decades or even a century away. The first goal is already partially accomplished. I believe that his work toward these two goals perfectly fits the definitions 3a, 5a, and 5b from Merriam-Webster.

I would like to point out that public opinion about various branches of mathematics changes continuously and in a manner internal to the mathematics itself. An area of mathematics may be (or may seem to be) completely exhausted; whatever is important in it, is relegated to textbooks, and a research in it wouldn’t be very valued. Somebody may prove a startling result by an unexpected new method; until the power of this method is exhausted, using it will be a very fashionable and valuable direction of research. This is just two examples.

In contrast with this, T. Gowers relies on ideological arguments, and, as one may guess, on his personal influence (note that most of the mathematical politics is done behind the closed doors and leaves no records whatsoever). In 2000, T. Gowers published two essays: “Two cultures in mathematics” in a highly popular collection of articles “Mathematics: Frontiers and Perspectives” (AMS, 2000), and “Rough structures and classification” in a special issue “GAFA Vision” of purely research journal “Geometric and functional analysis”.

The first essay, brilliantly written, put forward a startling thesis of the existence of two different cultures in mathematics, which I will call the mainstream and the Hungarian cultures for short. Most mathematicians are of the opinion that (pure) mathematics is a highly unified subject without any significant division in “cultures”. The mainstream culture is nothing else as the most successful part of mathematics in the century immediately preceding the publication of the “Two cultures” essay. It encompasses almost all interesting mathematics of the modern times. The Hungarian culture is a very specific and fairly elementary (this does not mean easy) sort of mathematics, having its roots in the work of Paul Erdös.

The innocently titled “GAFA Visions” essay has as it central and most accessible part a section called “Will Mathematics Exist in 2099?” It outlines a scenario eventually leading to a replacement mathematicians by computers. The section ends by the following prediction, already quoted in this blog.

"In the end, the work of the mathematician would be simply to learn how to use theorem-proving machines effectively and to find interesting applications for them. This would be a valuable skill, but would hardly be pure mathematics as we know it today.”

All arguments used to support the feasibility of this scenario are borrowed from the Hungarian culture. On the one hand, this is quite natural, because this is the area of expertise of Gowers. But then the conclusion should be “The work in the Hungarian culture would be simply to learn how to use Hungarian-theorems proving machines effectively”. This would eliminate the Hungarian culture, if it indeed exists, from mathematics, but will not eliminate pure mathematics.

This second project does not seem to be very realistic unless the mathematical community will radically change its preferences from favoring the mainstream mathematics to favoring the Hungarian one. And indeed, it seems that Gowers working simultaneously on both projects. He advocates Hungarian mathematics in his numerous lectures all over the world. He suddenly appears as the main lecturer on such occasions as the announcement of the Clay Institute million dollars prizes. It was a shock when he gave the main lecture about Milnor’s work at the occasion of the award of Abel prize to Milnor. Normally, such lectures are given by an expert in an area close to the area of the person honored. Gowers is in no way an expert in any of the numerous areas Milnor worked in. Moreover, he hardly had any understanding of the most famous results of Milnor; in fact, he consulted online (in a slightly veiled form at Mathoverflow.org) about some key aspects of this result. This public appearance is highly valuable for elevating the status of the Hungarian mathematics: a prominent representative of the last presents to the public some of the highest achievements of the mainstream mathematics.

The next year Gowers played the same role at the Abel prize award ceremony again. This time he spoke about his area of expertise: the award was given to a representative of Hungarian mathematics, namely, to E. Szemerédi. Be a presenter of a laureate work two year in a row is also highly unusual (I am not aware about any other similar case in mathematics) and is hardly possible without behind the closed doors politics. The very fact of awarding Abel prize to E. Szemerédi could be only the result of complicated political maneuvers. E. Szemerédi is a good and interesting mathematician, but not an extraordinary one. There are literally hundreds of better mathematicians. The award of the Abel prize to him is not an indicator of how good mathematician he is; it informs the mathematical community that the system of values of the mathematical establishment has changed.

How it could happen without politics that Gowers was speaking about the work of Milnor at the last year Abel prize ceremony? Gowers speaking about the work of Szemerédi is quite natural, but Gowers speaking about the work of Milnor (and preparing this presentation with the help of Mathoverflow) is quite bizarre. It is obvious that Gowers is the most qualified person in the world to speak about the works of Szemeredi, but there are thousands of mathematicians more qualified to speak about Milnor’s work.


Next post: The Politics of Timothy Gowers. 3.