Alexandre Grothendieck passed away yesterday, November 13, 2014

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Alexandre Grothendieck, the greatest mathematician for the twenties century, passed away on November 13, 2014 at the Saint-Girons hospital (Ariège) near the village Lasserre.

Alexandre Grothendieck spent about the last 24 years of his life in this village in Pyrenees range of mountains in a self-imposed retirement avoiding all contacts with the outside world and the mathematical community.
He had good reasons for this, but till now the mathematical community does not want to listen, or, rather, to read his extensive partially autobiographical, partially philosophical texts.

Alexandre Grothendieck, with help of his pupils, collaborators, and admires, completely transformed mathematics. His best known contribution is the proof of most of the Andre Weil conjectures (with the last step done by his pupil Pierre Deligne). Much more important is his transformation of the algebraic geometry from relatively obscure branch of mathematics to its central part. Even more important is his most intangible contribution, the concept known as th "rising sea", the idea that every mathematical problem should be immersed in a sufficiently abstract theory, which will made the solution trivial. This theory should be, in a sense, trivial too - it should not involve any tricks or convoluted arguments. This was a drastic departure from the mathematical analysis, the central branch of mathematics at the time, which was dominated by proofs demonstrating not so much the vision, but the "executive power" of the authors (the concept introduced by G. Hardy, who valued the executive power most). These ideas are still far from being internalized or even understood by the mathematical community.

Despite his tremendous influence, surpassing by a large margin the influence of any mathematician after David Hilbert, Alexandre Grothendieck was at least about 100 years ahead of his time.

His integrity and his concern about the perils people put each other into are hardly matched by any other contemporary scientist. He did not succeed much in this respect, apparently because his concerns only appeared to be left wing politics, but in fact were not of political nature.

With Alexandre Grothendieck passing away we lost the last living giant in mathematics.

Here is a link to a memorial article Alexandre Grothendieck, le plus grand mathématicien du XXe siècle, est mort in Le Monde, France (in French).



Next post: Where one can find an autobiography of Alexander Grothendieck? Part 1.

Is algebraic geometry applied or pure mathematics?

Previous post: About some ways to work in mathematics.

From a comment by Tamas Gabal:

“This division into 'pure' and 'applied' mathematics is real, as it is understood and awkwardly enforced by the math departments in the US. How is algebraic geometry not 'applied' when so much of its development is motivated by theoretical physics?”

Of course, the division into the pure and applied mathematics is real. They are two rather different types of human activity in every respect (including the role of the “problems”). Contrary to what you think, it is hardly reflected in the structure of US universities. Both pure and applied mathematics belong to the same department (with few exceptions). This allows the university administrators to freely convert positions in the pure mathematics into positions in applied mathematics. They never do the opposite conversion.

Algebraic geometry is not applied. You will be not able to fool by such statement any dean or provost. I am surprised that it is, apparently, not obvious anymore. Here are some reasons.

1. First of all, the part of theoretical physics in which algebraic geometry is relevant is itself as pure as pure mathematics. It deals mostly with theories which cannot be tested experimentally: the required conditions existed only in the first 3 second after the Big Bang and, probably, only much earlier. The motivation for these theories is more or less purely esthetical, like in pure mathematics. Clearly, these theories are of no use in the real life.

2. Being motivated by outside questions does not turn any branch of mathematics into an applied branch. Almost all branches of mathematics started from some questions outside of it. To qualify as applied, a theory should be really applied to some outside problems. By the way, this is the main problem with what administrators call “applied mathematics”. While all “applied mathematicians” refer to applications as a motivation of their work, their results are nearly always useless. Moreover, usually they are predictably useless. In contrast, pure mathematicians cannot justify their research by applications, but their results eventually turn out to be very useful.

3. Algebraic geometry was developed as a part of pure mathematics with no outside motivation. What happens when it interacts with theoretical physics? The standard pattern over the last 30-40 years is the following. Physicists use they standard mode of reasoning to state, usually not precisely, some mathematical conjectures. The main tool of physicists not available to mathematicians is the Feynman integral. Then mathematicians prove these conjectures using already available tools from pure mathematics, and they do this surprisingly fast. Sometimes a proof is obtained before the conjecture is published. About 25 years ago I.M. Singer (of the Atiyah-Singer theorem fame) wrote an outline of what, he hoped, will result from the interaction of mathematics with the theoretical physics in the near future. In one phrase, one may say that he hoped for infinitely-dimensional geometry as nice and efficient as the finitely-dimensional geometry is. This would be a sort of replacement for the Feynman integral. Well, his hopes did not materialize. The conjectures suggested by physicists are still being proved by finitely-dimensional means; physics did not suggested any way even to make precise what kind of such infinitely-dimensional geometry is desired, and there is no interesting or useful genuinely infinitely-dimensional geometry. By “genuinely” I mean “not being essentially/morally equivalent to a unified sequence of finitely dimensional theories or theorems”.

To sum up, nothing dramatic resulted from the interaction of algebraic geometry and theoretical physics. I don not mean that nothing good resulted. In mathematics this interaction resulted in some quite interesting theorems and theories. It did not change the landscape completely, as Grothendieck’s ideas did, but it made it richer. As of physics, the question is still open. More and more people are taking the position that these untestable theories are completely irrelevant to the real world (and hence are not physics at all). There are no applications, and hence the whole activity cannot be considered as an applied one.


Next post: The role of the problems.

Did J. Lurie solved any big problem?

Previous post: Guessing who will get Fields medals - Some history and 2014.

Tamas Gabal asked the following question.

I heard a criticism of Lurie's work, that it does not contain startling new ideas, complete solutions of important problems, even new conjectures. That he is simply rewriting old ideas in a new language. I am very far from this area, and I find it a little disturbing that only the ultimate experts speak highly of his work. Even people in related areas can not usually give specific examples of his greatness. I understand that his objectives may be much more long-term, but I would still like to hear some response to these criticisms.

Short answer: I don't care. Here is a long answer.

Well, this is the reason why my opinion about Lurie is somewhat conditional. As I already said, if an impartial committee confirms the significance of Lurie’s work, it will remove my doubts and, very likely, will stimulate me to study his work in depth. It is much harder to predict what will be the influence of the actual committee. Perhaps, I will try to learn his work in any case. If he will not get the medal, then in the hope to make sure that the committee is wrong.

I planned to discuss many peculiarities of mathematical prizes in another post, but one of these peculiarities ought to be mentioned now. Most of mathematical prizes go to people who solved some “important problems”. In fact, most of them go to people who made the last step in solving a problem. There is recent and famous example at hand: the Clay $1,000,000.00 prize was awarded to Perelman alone. But the method was designed by R. Hamilton, who did a huge amount of work, but wasn’t able to made “the last step”. Perhaps, just because of age. As Perelman said to a Russian news agency, he declined the prize because in his opinion Hamilton’s work is no less important than his own, and Hamilton deserves the prize no less than him. It seems that this reason still not known widely enough. To the best of my knowledge, it was not included in any press-release of the Clay Institute. The Clay Institute scheduled the award ceremony like they knew nothing, and then held the ceremony as planned. Except Grisha Perelman wasn’t present, and he did not accept the prize in any sense.

So, the prizes go to mathematicians who did the last step in the solution of a recognized problem. The mathematicians building the theories on which these solutions are based almost never get Fields medals. Their chances are more significant when prize is a prize for the life-time contribution (as is the case with the Abel prize). There are few exceptions.

First of all, A. Grothendieck is an exception. He proved part of the Weil conjectures, but not the most important one (later proved by P. Deligne). One of the Weil conjectures (the basic one) was independently proved by B. Dwork, by a completely different and independent method, and published earlier (by the way, this is fairly accessible and extremely beautiful piece of work). The report of J. Dieudonne at the 1966 Congress outlines a huge theory, to a big extent still not written down then. It includes some theorems, like the Grothendieck-Riemann-Roch theorem, but: (i) GRR theorem does not solve any established problem, it is a radically new type of a statement; (ii) Grothendieck did not published his proof, being of the opinion that the proof is not good enough (an exposition was published by Borel and Serre); (iii) it is just a byproduct of his new way of thinking.

D. Quillen (Fields medal 1978) did solve some problems, but his main achievement is a solution of a very unusual problem: to give a good definition of so-called higher algebraic K-functors. It is a theory. Moreover, there are other solutions. Eventually, it turns out that they all provide equivalent definitions. But Quillen’s definitions (actually, he suggested two) are much better than others.

So, I do not care much if Lurie solved some “important problems” or not. Moreover, in the current situation I rather prefer that he did not solved any well-known problems, if he will get a Fields medal. The contrast with the Hungarian combinatorics, which is concentrated on statements and problems, will make the mathematics healthier.

Problems are very misleading. Often they achieve their status not because they are really important, but because a prize was associated with them (Fermat Last Theorem), or they were posed by a famous mathematicians. An example of the last situation is nothing else but the Poincaré Conjecture – in fact, Poincaré did not stated it as a conjecture, he just mentioned that “it would be interesting to know the answer to the following question”. It is not particularly important by itself. It claims that one difficult to verify property (being homeomorphic to a 3-sphere) is equivalent to another difficult to verify property (having trivial fundamental group). In practice, if you know that the fundamental group is trivial, you know also that your manifold is a 3-sphere.

Next post: New ideas.

D. Zeilberger's Opinions 1 and 62

Previous post: Combinatorics is not a new way of looking at mathematics.

While this is a reply to a comment by  Shubhendu Trivedi in Gowers's blog, I hope that following is interetisting independently of the discussion there.

Opinion 1. Zeilberger admits there that he has no idea about the methods used even in his examples (the 4th paragraph).

He is correct that Jones polynomial is to a big extent a combinatorial gadget. Probably, he is not aware that this gadget applies to topology only if you have a purely topological theorem at your disposal (proved by Reidemeister in 1930s, it remains to be a non-trivial theorem). He may be not aware also of the fact that Jones polynomial did not led to solution of any problem of interest to topologists at the time. The proof of the so-called Tait conjecture was highly publicized, and many people believe that this was an important conjecture. Fortunately, there is a document proving that this is not the case. Namely, R. Kirby with the help of many other topologists compiled around 1980 a list of problems in topology. About 15 years later he published an updated and expanded version. Both editions consist of several parts, one of which is devoted to problems in knot theory. Tait conjecture is about knots and it is not in the 1980 list (by time Kirby started to prepare the new expanded list, it was already proved). Nobody was interested in it, and its solution has no applications.

Eventually, the theory of Jones polynomial and its generalizations turned into an independent self-contained field, desperately searching for connections with other branches of mathematics or at least with topology itself.

But D. Zeilberger should be aware that the Tutte polynomial belongs to the conceptual mathematics. It is one of the precursors of one of the main ideas of Grothendieck, namely, of K-theory. There is no reasons to think that Grothendieck was aware of Tutte's work, but Tutte polynomial is still an essentially a K-theoretic construction.

The Seiberg-Witten ideas have nothing to do with combinatorics. The Seiberg-Witten invariants are based on topology and some advanced parts of the theory of nonlinear PDE. In the last decade some attempts to get rid of PDE in this theory were partially successful. They involve some rather combinatorics-like looking pictures. I wonder if Zeilberger wrote anything about this. But the situation is essentially the same as with the Tutte polynomial. These quite remarkable attempts are inspired, not always directly, by such abstract ideas as 2-categories, for example. Note that the category theory is the most abstract part of mathematics, except, may be, modern set theory (which is a field in which only very few mathematicians are working).

Opinion 62. First, the factual mistakes.

Grothendieck did not dislike other sciences. In particular, at the age of approximately 42-46 he developed a serious interest in biology. Ironically, in the same paragraph Zeilberger commends I.M. Gelfand for his interest in biology.

Major applications of the algebraic geometry were not initiated by the “Russian” school, but the soviet mathematicians indeed embraced this field very enthusiastically. And initial applications did not involve any Grothendieck-style algebraic geometry.

More important is the fact that Zeilberger’s opinions are self-contradicting. He dislikes the abstract (in fact, the conceptual) mathematics, and at the same time praises the “Russian” school for applications of exactly the same abstract conceptual methods.

Zeilberger writes: “Grothendieck was a loner, and hardly collaborated”. Does he really knows at least a little about Grothendieck and his work? Grothendieck’s rebuilding of algebraic geometry in an abstract conceptual framework was a highly collaborative enterprise. He has almost no papers in algebraic geometry published by him alone. The foundational text EGA, Elements of Algebraic Geometry, has Grothendieck and Dieudonne as authors (in this order, violating the tradition to list the authors of mathematical papers in alphabetic order) and was written by Dieudonne alone. More advanced things were published as SGA, Seminar on Algebraic Geometry, and most of this series of Springer Lecture Notes in Mathematics Volumes is authored by Grothendieck and various collaborators. Some present his ideas, but don’t have him as an author. One of them is written by P. Deligne and authored by P. Deligne alone.

Zeilberger has no idea about what kind of youth was given to Grothendieck and presents some (insulting, I would say) conjectures about it. Grothendieck was always concerned with injustice done to other people, in particular within mathematics. His elevated sense of (in)justice eventually led him to (fairly misguided, I believe, but sincere and well-intentioned) political activity. He was initially encouraged by colleagues, who abandoned him when this enterprise started to require more than a lip service.

The phrase “...was already kicked out of high-school (for political reasons), so could focus all his rebellious energy on innovative math” is obviously absurd to everyone even superficially familiar with the history of the USSR. If someone was persecuted on political grounds, then (he could by summarily executed, but at least) any mathematical or other scientific activity would be impossible for him for life. There would be no ways to be a professor of Moscow State University, or taking part in the soviet atomic-nuclear project.

Surely, Gelfand said something like Zeilberger writes about the future of combinatorics. I never was at the Gelfand seminar, neither in Moscow, nor in Rutgers. But there are his publications, from which one can get the idea what kind of combinatorics Gelfand was interested in. Would Zeilberger attempted to read any of these papers, he would hardly see there even a trace of what is so dear to him. All works of Gelfand are highly conceptual.

Finally, it is worth to mention that Gelfand always wanted to be the one who determines the fashion, not the one who follows it. Of course, I see nothing wrong with it. In the late 60ies he regretted that he missed the emergence of a new field: algebraic and differential topology. He attempted to rectify this by two series of papers (with coauthors, by this time he did not published anything under only his name), one about cohomology of infinitely dimensional Lie algebras, another about a (conjectural) combinatorial definition of  Pontrjagin classes (a basic notion in topology). It is very instructive to see what was a “combinatorial definition” for I.M. Gelfand.

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Combinatorics is not a new way of looking at mathematics

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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.

Who writes about big questions?

Previous post: Slavic soul? The interest in big questions about mathematics.


Reply to a comment to the pervious post.

Well, your question makes it very tempting to write a long essay about the current state of mathematics. But this is not really needed.

I mentioned the most famous people; I hope that all names are immediately recognizable. Our contemporaries cannot be so famous just because they are our contemporaries. Let me try, but first I would like to say a few words about Thurston.

Yes, his essay is not quite about this topic. But there are hints, and his reply at Mathoverflow, which is reproduced in this blog is worth of 100s pages of other writers. A. Weil did not write a book on philosophy of mathematics, he just mentioned such issues here and there, and we know a lot about his views from his Bourbaki collaborators. The article I quoted in my first posts was published in French in an obscure (at least for non-French mathematicians) place. The “Monthly” translation expanded the audience, but the translation appears to be a not very good one. My point is that he wasn’t concerned much about dissemination these ideas. It looks like Thurston was more concerned about his ideas.

The most obvious example is, of course, T. Gowers. He wrote two essays, the one about “two cultures”, the other about replacing mathematicians by computers (this is, in fact, a section in his GAFA Visions paper), and he writes about such things in his blog. I don’t like his ideas, but if somebody outlines a project of replacing mathematicians by computers and offers a justification for such a project, he is definitely writing about the larger place of mathematics in the world in the most dramatic way: do we need mathematics or not? His answer is “no, we don’t”. His writings are definitely related to his own work: all examples are taken from his corner of mathematics.

Even the n-Category Café itself is an example, and one of the persons running it, David Corfield, wrote a book about philosophy of mathematics (I only browsed through it but plan to read it; it seems to be quite interesting). There is Colin McLarty, who writes about the implications of Grothendieck’s way of thinking. Of course, there is an autobiographical text (or, rather, several texts) of Grothendieck himself, which nobody dares to publish for 30 years already. On a much less abstract level, there is Neal Koblitz, who wrote about the role of mathematics in the society and criticized (largely from a political perspective) the way the “help” is given to the developing countries (and wrote an autobiographical book).

On the other hand, one can easily speak about mathematicians with Soviet-Russian upbringing, but how many texts written by them and worth reading can you suggest? Manin is excluded. Borovik’s book did not impress me enough to read it. Is my impression wrong? Honestly, I don’t know, but it is the business of the author to attract readers.

Finally, and perhaps most importantly, nowadays mathematics has a quite respectable place in the society. This allows us to earn a living by doing mostly the things we like to do no matter what. It would be too dangerous to try to insert a controversy about the larger role of mathematics in the society. Almost nobody dares to say anything nontrivial about this. Paul Halmos once said that NSF grants or any other government financing is not needed, because it does not matter when a theorem is proved, tomorrow or in 300 years. I don’t know any other comparable statement.

Personally, I believe that the government financing had already damaged mathematics too much and should be eliminated or, at least, radically reformed. Looking at the whole society, I believe that the teaching of mathematics in high schools and to almost all college students (in the US) is a serious damage to the society. But, this gives us our jobs! And these are just the most obvious issues.




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