A comment from Timothy Gowers

Previous post: Self-revealing truths? - Part 1

Let me try to dispel the mystery about my views. I share your distaste for incomprehensible proofs that merely certify the truth of a statement...

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Self-revealing truths? - Part 1

Previous post: Mathematicians are human and want to be famous

There is an apparently unnoticed class of truths, which I call “self-revealing” - statements which may be trivial, may be unexpected, but which are reveal themselves as true after being stated. They may require some thinking over, but they do not require arguments.

Or, may be, there is no such thing as a universally self-revealing truth? May be different truths reveal themselves as such to different people? May be even they are true for one person and false for another?

Here are some observations about mathematics, which are self-revealing truths for me, but which, apparently, are not even true for many other mathematicians. I am hardly able to support them by arguments. Or, rather, I am hardly willing to do this before I will face a deep challenge to their validity. But I will try to present them as a coherent whole.

Mathematics is a human activity. 

A distinctive feature of mathematics is an attempt to rely only on infallible arguments. This feature is known as the mathematical rigor, as also the concept of a rigorous proof. In fact, a mathematician will tell you that a non-rigorous proof is a non-sense: if something is a proof in mathematics, then it is rigorous. 

If mathematics has any value, then only as a human activity. 

I expect that here I will be pointed to the applications of mathematics, which should be valued independently of source of the mathematics applied: humans, computer, prophets, aliens from the outer space, etc. I would like to point out that this argument deserves some consideration only because of a historical accident: the fact that some Ancient Greeks interested in geometry were also interested in rigorous arguments. If not them, we would, most likely, have only non-rigorous applied mathematics. The self-revealing truth here is the following.

Applications of mathematics do not require rigorous proofs to be useful. 

Let me illustrate this by few observations.

In applications usually one can ignore special cases, or even the cases perceived as anomalous. This is something that mathematicians do not allow to themselves.

It is not widely known that the works of S. Fefereman, H. Friedman, S. Simpson, and G. Takeuti, among others, showed that the part of mathematics which is indispensable for scientific applications can be reduced to the (Peano) arithmetic. The questions about real numbers and sets of real numbers, which fascinated and inspired mathematicians for at least the last couple of centuries, are irrelevant for applications.

The following should be a complete triviality. The mathematical rigor is not needed for applications. We freely use many tools without being convinced or even assured that they will never fail. We use our tools knowing for sure that they will fail, sometimes quite often. Bridges, cars, planes from time to time fail to function as expected. Anybody reading this text on the web is familiar with both software and hardware failures.

One of the most widespread current applications of mathematics is the use prime numbers for encryption. Thanks to the modern cryptography, very big prime numbers even have definite monetary value. But one needs only numbers which are prime with very high probability. They are as good as the numbers which were verified to be prime with full mathematical rigor. If a number sold as a prime number is actually not prime, then a transaction at Amazon.com may fail. But that’s not a problem, occasionally they do fail anyhow by other reasons.

The most distinctive feature of mathematics, the concept of a mathematical proof, is not needed for applications.

Next post: A comment from Timothy Gowers

Mathematicians are human and want to be famous

Previous post: Where one can find an autobiography of Alexander Grothendieck? Part 2


A draft of this post was written quite a while ago. It was intended to be an opening of a series of posts. These posts may be written soon, or may never be will be written. Anyhow, I decided to post it "as is". The original title of the post was the following.

"Mathematics is a human activity."

This used to be such a platitude that hardly anybody dared to write it down, at least is such a straightforward form. This is not the case anymore. Some of the most prominent mathematicians do no share this position, including two Fields medalists. Timothy Gowers advocates a program of replacing mathematicians by computers. Vladimir Voevodsky quit algebraic geometry and algebraic K-theory, at least for the time being, and working on a program of computer verification of proof. Admittedly, his goals are more modest: computer verification of proofs; finding proofs is still left to humans.

A highly publicized example is the claim that Thomas Hales proved the so-called Kepler’s conjecture. The “proof” depends on verification of a huge number of statements by computers. His proof is almost accepted. Apparently, the experts are not quite satisfied even with the part of his proof he wrote for humans. Being not satisfied by such an “almost acceptance”, he decided to produce a new proof. But he has no intention to present a proof which his fellow mathematicians will be able to comprehend to their satisfaction. He is guiding a big project which, he expects, will lead to a proof of Kepler’s conjecture verifiable by computers from the first principles. In other words, his answer to the difficulties of his fellow mathematicians in reading his article (published in “Annals of Mathematics” with a special preface by the editors) is to make the whole argument inaccessible to humans.

Of course, the computer-assisted solution of the 4-colors problem by K. Appel and W. Haken is well known even to the general public. This computer-assisted proof is old enough to judge its usefulness for mathematics. It eliminated a stimulating problem in the graph theory, and did not lead to any progress on related conjectures such as the Hadwiger conjecture, for example. Since the claimed theorem itself is useless even within the pure mathematics, the total impact is negative.

All these mathematicians are missing the main point: mathematics is a human activity. They miss it notwithstanding the fact that they themselves are humans, and experience feelings similar to humans working in less esoteric fields.

Voevodsky is the most open about his motives: he was frustrated by the difficulties he experienced in writing down his proof of the Bloch-Kato conjecture and even in convincing himself that his proof is correct. In the end, he convinced both himself and the others, but decided to look for a less painful way to justify his claims. Knowing him to some extent personally, I believe that the fact these claims are his claims is important for him. He would not care much about claims that a computer proved this or that. And this may be the reason why his program seems to be the most meaningful one.

I risk to suggest that both Th. Hales and W. Haken (the driving force of the Haken – Appel team) are (were) motivated by their quest for “immortality” in the sense of the novel “Immortality” by Milan Kundera. To put it simple, by the desire to be remembered as the ones who solved a famous problem, or, at least, a problem as famous as possible.

W. Haken devoted years to attempts to prove the Poincaré conjecture before turning his attention to the 4-colors problem. K. Appel was a computer scientist and, most likely, had a different motivation.

I do not think that a quest for this sort of immortality is a bad thing per se. But when it hurts somebody or something I am attached a lot, as I am attached to mathematics, I dare to disapprove.

Timothy Gowers remains a mystery. As he wrote, he is glad that the humans will not be replaced by computers during his lifetime, but hopes that humans will be eliminated from mathematics in few decades.




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Where one can find an autobiography of Alexander Grothendieck? Part 2

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


A few years ago Grothendieck himself complicated the matter a lot. Note that this happened decades after his texts were rejected by all publishers.

Grothendieck contacted one or two of his former students and demanded that his works published without his authorization were removed from circulation, including libraries. At the time an extensive work, devoted to typesetting in TeX and simultaneously correcting misprints and minor mistakes, and clarifying his works when possible, was underway. Most of the Grothendieck's works were not written by him, and were published either as joint papers with J. Dieudonne (who wrote them all, but is listed as the second author contrary to the mathematical habit to list authors in the alphabetical order), or as notes of the "Séminaire de Géométrie Algébrique du Bois Marie" by Grothendieck and many of his pupils. As all such seminar notes, they are far from being perfect, and they not only deserve to be carefully rewritten, they need to be rewritten. After Grothendieck’s request, this work was almost completely halted, and the rewritten, but not yet published parts were taken down from the web. The already published part of the work was clearly subjected to Grothendieck request, but nothing was done about this. It seems that at least some of the still available paper publications were soon sold out, but some other are still available. (My presentation of this story posted few hours ago wasn't quite correct; the above is the corrected version.)

Note that Grothendieck had both moral and legal rights to demand this at least with respect with the notes of the "Séminaire de Géométrie Algébrique du Bois Marie", his most important mathematical texts. His moral rights as an author are obvious. In addition, he was the copyright holder. Originally, almost all these texts were published by Springer, and the copyright, as usual, belonged to Springer. But at the end of the 1980-ies Springer returned the copyright to Grothendieck. So, legally, nobody can do anything with these texts without Grothendieck’s permission.

The people involved esteemed Grothendieck too much to openly go against his will. Presumably, some people continued to rewriting, but without posting their text on the web (and, of course, without publishing them in the conventional sense).

The situation with his autobiography is much simpler. While it is possible to argue that his discoveries do not belong to him - they belong to humanity, this is not the case with his autobiographical texts. They are like personal letters. They were never published. So, both the moral and the legal rights belonged to Grothendieck. Given the fact that people were very reluctant to do anything against his desire with his mathematical texts, they are even more careful with his personal texts.

Now, after Grothendieck passed away, both the moral and legal rights belongs to his surviving relatives. While the New York Times wrote that he has no known survivors, this seems to be incorrect, and his surviving relatives have no objection against circulation of at least his mathematical texts. Probably, the project of rewriting of the notes of the "Séminaire de Géométrie Algébrique du Bois Marie" will resume. Of course, the original notes are available in many copies.

It is less clear what will happen with his autobiography. Of course, there were many copies in circulation, and I doubt that everyone in possesion of such a copy, be it paper or electronic, destroyed it. If you are lucky, you may come across such a person or even find something on the web. I would very much appreciate any references.

In 1990-ies a Russian translation of the first two parts of Grothendieck's autobiography was published in a completely regular manner. If you read Russian, you should be able to easily find copies on the web. The Russian title is "Урожаи и посевы".

Note that most of mathematical text by Grothendieck and all non-mathematical are in French. While this seems to be a hardly serious obstruction in the case of mathematical papers, his autobiography is written in a rather poetic and sophisticated French. At least one person started to translate it in English, but this is a time consuming task, and he needs to earn a living. He needs funds. Probably, he needs also assurance that his work will be published in some way: will be made easily accessible.

At the same time, even a biography of Grothendieck, partially written by a well known and respected German mathematician W. Scharlau, turned out to be unpublishable in a regular way. The already completed parts are more or less self-published, and there is a need to fund an English translation. See Translation of Grothendieck Biography. The translation of the first part is available at Amazon as a book on demand: Winfried Scharlau, Who Is Alexander Grothendieck? Part 1: Anarchy. The German original of the 3rd part is also available on Amazon as a book on demand: Winfried Scharlau, Wer Ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualit T, Einsamkeit Eine Biographie Teil 3 (German Edition).



Next post: Mathematicians are human and want to be famous.

Where one can find an autobiography of Alexander Grothendieck? Part 1

Previous post: Alexandre Grothendieck passed away yesterday, November 13, 2014.


michal2602 asked this question in a comment to the previous post. The short reply would be "I have no idea". This post and the next one are devoted to a long reply.

I don't know, and by good reasons.

First of all, autobiographical and philosophical texts of Grothendieck were never published. They were offered (I am not sure that by Grothendieck himself) to some publishers in France, and everyone rejected the offer. I was told that in his autobiographical texts Grothendieck applied to his colleagues and his own students’ very high moral standards, and points out the violation of these standards. Moreover, sometimes he points out violation of the common standards of scientific ethics or even of the common decency standards. The problem is that he names the violators. And this is something that is quite risky (for the potential publisher) in France (or so I was told).

At the same time the mathematical community does not like such things at all (this is my observation). The mathematical community prefers not to investigate even the cases of nearly oblivious stealing of theorems or ideas (even when an investigation will clear the accused). If your theorem is stolen, you are better off if you do not tell about this in public (unless your proof was literally copy-pasted from your paper to a paper of somebody else).

Apparently, Americans are much more tolerant to the public discussion of any aspect of the life of all sorts of celebrities (the legal term is the “public person”). As is well known, the right to discuss this is codified in the First Amendment to the US Constitution and its Supreme Court interpretations. And why somebody in the US would care about an accusation of a member French Academy? It would be quite natural to translate the Grothendieck’s autobiography in English and to publish it. The American Mathematical Society is the most natural publisher for such a translation. This never happened. The American Mathematical Society considers Grothendieck’s autobiography to be just not interesting enough.

For me, all this is rather depressing. I used to think that the scientific community (including the mathematical one) is open and welcoming controversies. In my opinion, everything written by a mathematician of high enough caliber should be published (may be except wrong proof, but even some wrong proofs deserve to be published). Grothendieck’s caliber is much higher than necessary for this. If such a mathematician holds currently unacceptable opinion, let us argue about it. If she or he misunderstood something, or wasn’t well informed, let us point out at the mistake. But we should not silence people. We don’t have to publish all the rubbish people can produce, but if we deal with a genius, we cannot be certain that we can tell apart the rubbish from that we just don’t understand yet.


Next post: Where one can find an autobiography of Alexander Grothendieck? Part 2.

Alexandre Grothendieck passed away yesterday, November 13, 2014

Previous post: And who actually got Fields medals?


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.

And who actually got Fields medals?

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

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

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

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

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

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


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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?

About expository writing: a reply to posic

Previous post: Graduate level textbooks: A list - the second part


In the post Graduate level textbooks I I mentioned an advice given to me by a colleague many years ago:
"Do not write any books until you retire". posic commented on this:

"Do not write any books until you retire"?! One is tempted to generalize to "do not do any mathematics until you retire". Or, indeed, to "do not do anything you find interesting, important or meaningful until you retire"...

Gone are the days when Gian-Carlo Rota wrote "You are most likely to be remembered for your expository work" as one of his famous "Ten lessons I wish I had been taught". Not that I so much like this motivation, that is one's desire to have oneself remembered at any expense, but compared to people doing mathematics from the main motivation of getting tenure, grants, etc., it was, at least, leaving ground for some cautious hope. Presently I do not see any.

I am sorry for the long delay with a reply. Here are some thoughts.

The advice of my colleague does not admit such generalizations. He based it on the opposite grounds: he wanted me to do something more interesting than writing books.

He made a couple of common mistakes. First, he has no way to know what is interesting to other people, including myself. A lot of people do find writing expository works (at any level, from elementary school to the current research) to be very interesting. Actually, I do. At the same time, many mathematicians complain about lack of necessary expository writings. Some direction of research died because the discoverers are not able to write in an understandable manner, and others were discouraged to write expositions. At the same time, writing down some ideas is a creative work at a level higher than most of “Annals of Mathematics” papers.

Second, he followed a prejudice common at least in the US: expository writing is a second-rate activity compared to proving theorems. This prejudice is so strong that proving “empty” theorems is valued more than excellent expository writing. Apparently, this is a result of external with respect to mathematics influences. The main among them is the government funding of pure mathematics. There is essentially only one agency in the US providing some funds for pure mathematics, namely, the NSF. The role of few private institutions is negligible. It is not surprising that NSF has its own preferences, and the pure mathematics is not its main concern. Moreover, it is very likely that NSF is even not allowed by law to fund expository writing (I did not attempted to check this).

G.-C. Rota is right. He almost always right, especially if you at least try to read between the lines. Actually, the most cited (and by a wide margin) work of the mentioned colleague is a purely expository short monograph. So, he does not put his money where his mouth is.

Actually, I am not inclined to read G.-C. Rota so literally. He is a too sophisticated thinker for this. Whatever he says, he says it with a tongue in cheek. He wanted to encourage expository writing. The motivation he offered isn’t really the fame. It is the usefulness. You will be remembered most for things most useful for other people. For many expository writing will be much more useful than publishing a dozen of “research” papers.

I think that it will come as no surprise to you that the government agencies, supposedly to work on behalf of the people, demand a lot of work hardly useful to anybody, and do not support really useful (at least to some people) activities. I also believe that only few other mathematicians will agree.

Doing mathematics for getting tenure or its equivalent is essentially doing mathematics for having an opportunity to do mathematics. There are no other ways. If you know a way to do mathematics without an equivalent of a tenured academic position in the US, please, tell me. I do have tenure, but I am quite interested.

This is not so with "grants, etc.", especially if you have tenure. Working for grants is a sort of corruption. Unfortunately, it is so widespread. Well, some people, for example G.W. Mackey, predicted this at the very beginning of the government funding. They turned out to be correct.

G.-C. Rote wrote these words quite a while ago. Things did not improve since then. The expository writing is valued even less than at the time. Nobody cares if he/she or you will be remembered 100 years from now, or if a current paper will be remembered 10 years from now. Everything is tailored for the medicine and biology. Reportedly, almost no papers there are remembered or cited after 2 years. Anyhow, the infamous impact factor of a journal takes into account only the citations during the first 2 years after the publication. The journals are judged by their impact factor, the papers are judged by the journals where they are published, and academics are judged by the quantity (in the number of papers, not pages) and the "quality" of their publications.

Apparently, mathematicians are content with the current situation and are afraid of any changes more than cosmetic ones. Is there a hope?


Next post: To appear

Graduate level textbooks: A list - the second part

Previous post: Graduate level textbooks: A list - the first part


N. Koblitz, p-adic number, p-adic analysis, and zeta-functions. GTM. Perfect in every respect.

N. Koblitz, Other books. It seems that all of them are also excellent, but I am less familiar with them (the previous one I read from cover to cover).

K. Kunen, Set theory: an introduction to independence proofs. This is the best exposition of P. Cohen’s method of proving the independence of continuum-hypothesis (there is no other method). I do not think anymore that this independence is such a big deal as people used to think and many still think. The reason is that I do not attribute to this theorem any philosophical significance, and this is because I know its proof, which I learned from Kunen’s book. But Cohen’s proof is very beautiful and subtle. I learned this from Kunen’s book too. All this beauty and subtlety are missing from popular expositions, even from ones written for mathematicians.

I. Lakatos, Proof and refutations. This is a rather unusual book devoted to the philosophy of mathematics. Definitely not a textbook, but highly recommended. Brilliantly written.

S. Lang, Algebra. The last edition is more than two times longer than the first. A lot of people hate this book as too abstract. They miss the point: the goal of the book is to teach to think in abstract terms. GTM

S. Lang, An introduction to algebraic and abelian functions. GTM

S. Lang, Other books. The collection of Lang’s books is huge and uneven. I will not suggest reading his undergraduate calculus textbooks, but his lectures for high school students are excellent. Many people don’t like Lang’s books without realizing that to a big extend Lang defined the modern style of an advanced mathematics textbooks, and that many books they like are either written in this style, or are just watered down versions of books written in this style (or even of books written by Lang himself).

O. Lehto, Univalent functions and Teichmüller spaces. GTM

G. Mackey, Lectures on mathematical foundations of quantum mechanics.

S. MacLane, Homology. This is a classic written with perfect timing: when a new branch of mathematics (homological algebra) just turned into a mature subject.

S. MacLane, Categories for the working mathematician. GTM

Yu.I. Manin, A course in mathematical logic for mathematicians. It is worthwhile even just to browse this book looking for general remarks. There are a lot of deep insights hidden in it. GTM

Yu.I. Manin. Other books, if you mastered the prerequisites.

W. Massey. Algebraic topology. An introduction. Later versions include homology theory. My recommendation is only for the fundamental groups part. GTM

J.W. Milnor, Morse theory.

J.W. Milnor, Topology from the differential viewpoint.

J.W. Milnor, An introduction to algebraic K-theory.

J.W. Milnor. All other books by Milnor are also exceptionally good with the only possible exception of the book about h-cobordism theorem (this one is really a long research-expository paper).

D. Mumford, Algebraic geometry. Complex projective varieties. One of the best books in mathematics I ever read.

D. Mumford, The red book of varieties and schemes. Probably, the best introduction to schemes.

D. Mumford, Curves and their Jacobians. These lecture notes cannot serve as a textbook, there are no complete proofs, but there is a wealth of insights and ideas; the exposition is masterful. These notes are included into the last Springer edition of The red book of varieties and schemes.

D. Mumford, Lectures on theta-functions I, II, III.

D. Mumford, Other writings. Everything (including research papers) written by Mumford the algebraic geometer is great if one has the required prerequisites. Unfortunately, he left the field and the pure mathematics in general in early 1980ies.

R. Narsimhan, Analysis on real and complex manifolds.

D. Ramakrishnan, R.J. Valenza, Fourier analysis on number fields. GTM

Elmer G. Rees, Notes on geometry. UTM (Springer Undergraduate Texts in Mathematics)

J. Rotman, Homological algebra. The first edition (Academic Press) is shorter and better than the second one (Springer). The first edition is a gem. The second edition contains much more material, which is at the same time a plus and a minus.

W. Rudin, Principles of mathematical analysis. I learned the basics of the mathematical analysis from this book within a month. This month was fairly horrible in almost all other respects.

W. Rudin, Functional analysis.

W. Rudin, Real and complex analysis.

W. Rudin, Fourier analysis on groups.

C. Rourke, B. Sanderson, Introduction to piecewise-linear topology. The book is perfect, but field is out of fashion. The reasons for the latter are not internal to the field; they are the same as in the fashion industry.

J.-P. Serre, Lie algebras.

J.-P. Serre, Lie groups.

J.-P. Serre, A course in arithmetic.

J.-P. Serre, Linear representations of finite groups.

J.-P. Serre, Trees. Perfect.

J.-P. Serre, Everything else, if you mastered the prerequisites.

I.R. Shafarevich, Basic of algebraic geometry, V. 1, 2. The best introduction to the algebraic geometry, but it is too slow if you are planning to be an algebraic geometer.

M.A. Shubin, Pseudo-differential operators and spectral theory.

E. Stein, Singular integrals and differential properties of functions.

E. Stein and Rami Shakarchi, 4 volumes of “Princeton Lectures in Analysis”. I did not read them, but I am sure that they are very good.

J.-P. Tignol, Galois' Theory of Algebraic Equations.

R. Wells, Differential analysis on complex manifolds. Reprinted 2008. GTM

F.W. Warner, Foundations of differentiable manifold and Lie groups. GTM

H-h. Wu, The Equidistribution Theory of Holomorphic Curves. This is a fairly old book and at the same time the last book I read from cover to cover (about two or three years ago). It is brilliant. Don’t be scared by long computations, especially in the last chapter: the author presents them in a way which shows their inner working.

Wu's book completes this list.

Next post: About expository writing: a reply to posic

Graduate level textbooks: A list - the first part

Previous post: Graduate level textbooks II


The following list includes only the books which I read from cover to cover or from which I read at least some significant part (with a couple of exceptions); the books which I just used in my work are not included, no matter how useful they were.

This list includes almost no recent titles; I am planning to compile a list of more recent titles later. There are several reasons for this. First, recent books did not pass the test of time yet. Second, by now I rarely need to read a textbook; my education was completed quite a while ago. Still, I am always happy to learn new things if there is an accessible way to do this. Unfortunately, for many things which (or about which) I would be very happy to learn, there are no expository texts at all, not to say about textbooks. In the ancient times (say, in 1960ies) people wrote excellent expositions accessible to non-experts within only few years after a new theorem or theory appeared. Apparently, this is not the case anymore. I see two main reasons for this. First, nowadays young people are required to publish several papers a year; they don’t have time to write a book. The other reason is the bizarre way in which the internet (and the new technology of printing books on demand) influenced the mathematical publishing. Whatever the reason is, much more good books in pure mathematics were published just 5 years ago.

The main factor determining if any book is good or bad is its author. Therefore, the other books by an author of a book included in the list deserve attentions. Occasionally, I mention this explicitly.

“GTM” means that the book was published or reprinted in the Springer series “Graduate Texts in Mathematics”.


L. Ahlfors, Lectures on quasi-conformal maps. Recently reprinted by the AMS.

V.I. Arnold, Mathematical methods of the classical mechanics. GTM

V.I. Arnold, Other books. Arnold style is far from being polished, and he inserts here and there many of his non-standard opinions. You don’t have to agree with his opinions, but it would be wrong to dismiss any of them outright. The value of his books lies in their personal style, not in giving the best expositions of standard topics.

W. Arveson, A short course on spectral theory. GTM

M. Atiyah, Lectures on K-theory. The proof of the Bott periodicity is not the best one and is fairly cumbersome. I suggest not spending much time on it.

M. Atiyah, I.G. Macdonald, An introduction to commutative algebra.

B. Bollobas, Modern graph theory, the last edition. For an outsider like me, it is written rather unevenly: some topics are presented very clearly and with all the details; some other topics are presented in a too condensed manner. GTM

K. Brown, Cohomology of groups. GTM

T. Bröcker, L. Lander, Differentiable germs and catastrophes. The topic is out of fashion, but this happened by external to it reasons and it still has a lot of potential.

T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups. GTM

N. Bourbaki, Lie groups and Lie algebras. (Chapters that are needed.)

N. Bourbaki, Commutative algebra. (Chapters that are needed.)

H. Clemens, A scrapbook of the complex curves theory. Recently reprinted by the AMS.

H. Edwards, Galois Theory. GTM

R.E. Edwards, Fourier series, A modern introduction. V. 1, 2. GTM

J.-P. Escofier, Galois Theory. GTM

R. Goldblatt, Topoi, the categorical analysis of logic.

I. Herstein, Noncommutative rings.

R. Hartshorne, Foundations of projective geometry, vii, 167 p. This one is elementary and recommended to be read before the basics of abstract algebra are learned.

R. Hartshorne, Algebraic geometry. GTM. Actually, this one is very good, but is not one of my favorites. This book has the reputation of being a must for entering the modern algebraic geometry, and this seems to be indeed the case. This is the reason for including it in the list.

Personally, I don’t like the style of this book. The core of the book is Chapters 2 and 3. They are much shorter than the corresponding parts of the EGA tract by Grothendieck-Dieudonne, but this is due mostly not to treating only less general situations, but to the fact that a huge amount of the material is presented as exercises without solutions, and in the main part of the text the author sometimes omits non-trivial arguments presented in details in EGA. Chapter 1 is a pre-Grothendieck introduction to algebraic geometry, and the last Chapters 4 and 5 illustrate the general theory of Chapters 2 and 3 by some classical applications.

K. Ireland, M. Rosen, A classical introduction to the modern number theory. GTM. Brilliant.

I. Kaplansky, Lie algebras and locally compact groups. This is actually two very short books under one cover. The first one is an introduction to Lie algebras, the second one is devoted to the solution of Hilbert’s fifths problem by Gleason and Montgomery-Zippin (it seems that a much longer book by Montgomery-Zippin is the only other exposition). I. Kaplansky always wrote with an ultimate elegance and his writing worth reading by this reason alone.

Continued in the next post.

Next post: Graduate level textbooks: A list - the second part

Graduate level textbooks II

Previous post: Graduate level textbooks I


I would like to start with something at least a little bit shocking.

My first list will consists of books by two excellent authors who wrote many books each. These two authors are as different as one can imagine. I will say also few words about a third author, who worked nearly three hunderd years ago. The books mentioned in this post are not suggested for the first reading. I do not suggest them for reading cover to cover either. I will turn to more conventional books in the next post.


N. Bourbaki, Commutative algebra

N. Bourbaki, Lie groups and Lie algebras

N. Bourbaki, Elements of the History of Mathematics

N. Bourbaki, Théories spectrales

N. Bourbaki, Variétés différentielles et analytiques: fascicule de résultats

N. Bourbaki, Algèbre, Chapitre 10. Algèbre homologique


I do not suggest the more foundational books by Bourbaki; they are not suitable as textbooks at all. Actually, none of them is written as a textbook or intended to be one. The books listed above are written at a fairly advanced level. It is expected that the reader already has a motivation to study a particular area. These books have a perfect selection and organization of the material; proofs are condensed, but there is no handwaving and all the details are there. The book on manifolds contains no proofs; it is only a resume of the theory. The Chapter about homological algebra, probably, should be considered as outdated. But it hardly possible to start with the modern form of homological algebra; in any case, there is no textbook doing this.


Harold M. Edwards, Riemann's zeta function. For experts or to be experts only.

Harold M. Edwards, Advanced Calculus, A Differential Forms Approach. This is how one should teach calculus. I am not sure that there is any real need to study or teach calculus, but this is another topic.

Harold M. Edwards, Fermat's last theorem: a genetic introduction to algebraic number theory. Brilliant. But nobody planning to be an expert in algebraic number theory will have time to learn from this book, following the historical development of algebraic number theory.

Harold M. Edwards, Divisor Theory. This book is accessible and interesting, but very specialized.

Harold M. Edwards, Galois theory. If you know something about the Galois theory, it would be very instructive to take a look at what Galois really did.

Harold M. Edwards, Linear Algebra. This book is written at the undergraduate level. As always, Edwards takes a non-standard approach. It is good, but I do not suggest studying the linear algebra from it. Actually, one should not study the linear algebra as a separate subject at all. The reason is the fact that there is no such branch of mathematics, and never was such a branch.

Harold M. Edwards, Essays in Constructive Mathematics. Don’t be misled by the title; it is not about what people usually call “constructive mathematics”. It is an introduction to algebraic number theory and algebraic curves which stresses the explicit results (so that you can actually compute something) and the historical perspective.

Harold M. Edwards, Higher arithmetic: an algorithmic introduction to number theory. The title says it all.


The books by Harold M. Edwards are distinguished, first of all, by putting the material in the historical perspective. He follows the motto “Learn from the masters” and makes the works of discoverers accessible to the modern readers. The modern expositions are usually not only streamlined, but also watered down a lot, sometimes to the extent of eliminating all content. His later books also stress the algorithmic and computational aspects. This does not suits my tastes well, but it gives a new perspective, and when I read such good writer (I do not mean that this is easy), I can not only forgive, but also appreciate this.

I must admit that I did not read even a single chapter from the last two books, but they are on my reading list.

The history of a mathematical theory is its main and usually the only motivation (may be after an initial impetus from the outside). By this reason it makes a lot of sense to read not only 40 years old research papers (for a mathematician there is nothing unusual in this), but even 200 years old books. The problem is that they are written in a language hardly understandable now, and, in addition, they are usually written in Latin (the mathematical Latin is not very difficult but still is a serious obstruction). L. Euler is an exception. His books (and papers) are written in a way accessible to a modern reader. They are written in a style quite different from the modern one: Euler very often explains how he or his predecessors reached the presented results, and these explanations are an integral part of the text. They are not relegated to appendices at the ends of chapters or sections. Also, he wrote about results he wasn’t able to prove, explaining why there are compelling reasons to think that they are true.

Perhaps, every modern mathematician will be surprised by how far his textbooks in calculus go. Of course, they consist of several fairly extensive volumes. Still, this is the calculus of his time, and I doubt that many contemporary mathematicians will be able to master his more advanced topics (which include questions considered now as parts of algebraic geometry).

The main problem with Euler’s writings is the lack of English translations. It seems that all his books are translated into all main European languages except English. Still, something is translated. If you have time, his books are highly recommended. In fact, they can be even used in undergraduate teaching, if you are inclined to teach something meaningful and accessible and your undergraduate director will allow you to do this.


Next post: Graduate level textbooks: A list - the first part

Graduate level textbooks I

Previous post: The role of the problems


Back in August Tamas Gabal asked me about my favorite graduate level textbooks in mathematics; later Ravi joined this request. I thought that the task will be very simple, but it turned out to be not. In addition, my teaching duties during the Fall term consumed much more energy than I could predict and even to imagine.

In this post I will try to explain why compiling a list of good books is so difficult. It is much easier to say from time to time “This book is great! You should read it.” Still, I will try to compile a list or lists of the books I like in the following post(s).

If one is looking for good collection of graduate level textbooks, there is no need to go further than the Springer series “Graduate Texts in Mathematics”. The books in the Springer “Universitext” series are more varied in their level (some are upper level undergraduate, others are research monographs), but one can find among them a lot of good textbooks. There is a more recent series “Graduate Studies in Mathematics” by the AMS. From my point of view, this series includes some excellent books, but is too varied both in terms of the level and in terms of quality. If you are looking for something on the border between an advanced graduate level textbook and a research monograph, the Cambridge University Press series “Cambridge Studies in Advanced Mathematics” is excellent. The bizarre economics and ideology of the modern scientific publishing resulted in the fact almost all good books in mathematics (including textbooks) is published by one of these 3 publishers: AMS, Springer, and Cambridge University Press. You will not miss much if will not go any further (but you will miss some book, certainly).

I cannot suggest a sequence of good books to study any sufficiently broad area, even not necessarily a sequence of my favorite books. If you want to be a research mathematician, you will have to learn a lot from bad books and badly written papers. It would do a lot of good for mathematics if afterwards you will write a good book about things you learned from badly written books and papers. Unfortunately, writing a book is not a really good idea at the early stages of the career of a mathematician nowadays. Expository writing is hardly valued. On the one hand, expository writing does not help to get grants and grants is the only thing valued by administrators at the level of deans and higher. It seems that the chairs of the mathematics departments started to follow this approach. Deans and chairs are the ones who have the last word in any hiring or promotion decision. Sometimes a mathematician is essentially forced to write a book in order to continue research. For example, the foundation of a theory may be absent from the literature, or some “known to everybody” results may require clarification. But this is rare.

Some freedom of what to do, in particular, the freedom to write books, arrives only with a tenured position. Still, a colleague of me gave me many years ago the following advise: “Do not write any books until you retire”. Right now I am not sure that any mathematical books will be written or used when I retire. I actually had abandoned a couple of projects because I don’t see any efficient and decent way to distribute mathematical books. I don’t think that charging $100.00 for a textbook is decent given that the cost of production is about $5.00—$20.00 per copy.

On the other hand, there is a lot of good textbook introducing into a particular sufficiently narrow branch of mathematics. It hardly make sense to list all of them. All this leads me to chosing “my favorite” as the guiding principle. And, after all, this is what Tamas Gabal asked me to do.


Next post: Graduate level textbooks II

The role of the problems

Previous post: Is algebraic geometry applied or pure mathematics?


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.

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.

About some ways to work in mathematics

Previous post: New ideas.


From a comment by Tamas Gabal:

“...you mentioned that the problems are often solved by methods developed for completely different purposes. This can be interpreted in two different ways. First - if you work on some problem, you should constantly look for ideas that may seem unrelated to apply to your problem. Second - focus entirely on the development of your ideas and look for problems that may seem unrelated to apply your ideas. I personally lean toward the latter, but your advice may be different.”

Both ways to work are possible. There are also other ways: for example, not to have any specific problem to solve. One should not suggest one way or another as the right one. You should work in the way which suits you more. Otherwise you are unlikely to succeed and you will miss most of the joy.

Actually, my statement did not suggest either of these approaches. Sometimes a problem is solved by discovering a connection between previously unrelated fields, and sometimes a problem is solved entirely within the context in was posed originally. You never know. And how one constantly looks for outside ideas? A useful idea may be hidden deep inside of some theory and invisible otherwise. Nobody studies the whole mathematics in the hope that this will help to solve a specific problem.

I think that it would be better not to think in terms of this alternative at all. You have a problem to solve, you work on it in all ways you can (most of approaches will fail – this is the unpleasant part of the profession), and that’s it. The advice would be to follow development in a sufficiently big chunk of mathematics. Do not limit yourself by, say, algebra (if your field is algebra). The division of mathematics into geometry, algebra, and analysis is quite outdated. Then you may suddenly learn about some idea which will help you.

Also, you do not need to have a problem to begin with. Usually a mathematician starts with a precisely stated problem, suggested by the Ph.D. advisor. But even this is not necessary.

My own way to work is very close to the way M. Atiyah described as his way of work in an interview published in “The Mathematical Intelligencer” in early 1980ies (of course, I do not claim that the achievements are comparable). This interview is highly recommended; it is also highly recommended by T. Gowers. I believe that I explained how I work to a friend (who asked a question similar to yours one) before I read this interview. Anyhow, I described my way to him as follows. I do not work on any specific problem, except of my own working conjectures. I am swimming in mathematics like in a sea or river and look around for interesting things (the river of mathematics carries much more stuff than a real river). Technically this means that I follow various sources informing about the current developments, including talks, I read papers, both current and old ones, and I learn some stuff from textbooks. An advanced graduate level textbook not in my area is my favorite type of books in mathematics. I am doing this because this is that I like to do, not because I want to solve a problem or need to publish 12 papers during next 3 years. From time to time I see something to which, I feel, I can contribute. From time to time I see some connections which were not noticed before.

My work in “my area” started in the following way. I was familiar with a very new theory, which I learned from the only available (till about 2-3 years ago!) source: a French seminar devoted to its exposition. The author never wrote down any details. Then a famous mathematician visited us and gave a talk about a new (not published yet) remarkable theorem of another mathematician (it seems to me that it is good when people speak not only about their own work). The proof used at a key point an outside “Theorem A” by still another mathematicians. The speaker outlined its proof in few phrases (most speakers would just quote Theorem A, so I was really lucky). Very soon I realized (may be the same day or even during the talk) that the above new theory allows at least partially transplant Theorem A in a completely different context following the outline from the talk. But there is a problem: the conclusion of Theorem A tells that you are either in a very nice generic situation, or in an exceptional situation. In my context there are obvious exceptions, but I had no idea if there are non-obvious exceptions, and how to approach any exceptions. So, I did not even started to work on any details. 2-3 years later a preprint arrived in the mail. It was sent to me by reasons not related at all with the above story; actually, I did not tell anybody about these ideas. The preprint contained exactly what I needed: a proof that there are only obvious exceptional cases (not mentioning Theorem A). Within a month I had a proof of an analogue of Theorem A (this proof was quickly replaced by a better one and I am not able to reproduce it). Naturally, I started to look around: what else can be done in my context. As it turned out, a lot. And the theory I learned from that French seminar is not needed for many interesting things.

Could all this be planned in advance following some advice of some experienced person? Certainly, not. But if you do like this style, my advice would be: work this way. You will be not able to predict when you will discover something interesting, but you will discover. If this style does not appeal to you, do not try.

Note that this style is opposite to the Gowers’s one. He starts with a problem. His belief that mathematics can be done by computers is based on a not quite explicit assumption that his is the only way, and he keeps a place for humans in his not-very-science-fiction at least at the beginning: humans are needed as the source of problems for computers. I don’t see any motivation for humans to supply computers with mathematical problems, but, apparently, Gowers does. More importantly, a part of mathematics which admits solutions of its problems by computers will very soon die out. Since the proofs will be produced and verified by computers, humans will have no source of inspiration (which is the proofs).


Next post: Is algebraic geometry applied or pure mathematics?

New ideas

Previous post: Did J. Lurie solved any big problem?


Tamas Gabal asked:

“Dear Sowa, in your own experience, how often genuinely new ideas appear in an active field of mathematics and how long are the periods in between when people digest and build theories around those ideas? What are the dynamics of progress in mathematics, and how various areas are different in this regard?”

Here is my partial reply.


This question requires a book-length answer; especially because it is not very precisely formulated. I will try to be shorter. :- )

First of all, what should be considered as genuinely new ideas? How new and original they are required to be? Even for such a fundamental notion as an integral there are different choices. At one end, there is only one new idea related to it, which predates the discovery of the mathematics itself. Namely, it is idea of the area. If we lower our requirements a little, there will be 3 other ideas, associated with the works or Archimedes, Lebesque, and hardly known by now works of Danjoy, Perron, and others. The Riemann integral is just a modern version of Archimedes and other Ancient Greek mathematician. The Danjoy integral generalizes the Lebesgue one and has some desirable properties which the Lebesgue integral has not. But it turned out to be a dead end without any applications to topics of general interest. I will stop my survey of the theory of integration here: there are many other contributions. The point is that if we lower our requirements further, then we have much more “genuinely new” ideas.

It would be much better to speak not about some vague levels of originality, but about areas of mathematics. Some ideas are new and important inside the theory of integration, but are of almost no interest for outsiders.

You asked about my personal experience. Are you asking about what my general knowledge tells me, or what happened in my own mathematical life? Even if you are asking about the latter, it is very hard to answer. At the highest level I contributed no new ideas. One may even say that nobody after Grothendieck did (although I personally believe that 2 or 3 other mathematicians did), so I am not ashamed. I am not inclined to classify my work as analysis, algebra, geometry, topology, etc. Formally, I am assigned to one of these boxes; but this only hurts me and my research. Still, there is a fairly narrow subfield of mathematics to which I contributed, probably, 2 or 3 ideas. According to A. Weil, if a mathematician had contributed 1 new idea, he is really exceptional; most of mathematicians do not contribute any new ideas. If a mathematician contributed 2 or 3 new ideas, he or she would be a great mathematician, according to A. Weil. By this reason, I wrote “2 or 3” not without a great hesitation. I do not overestimate myself. I wanted to illustrate what happens if the area is sufficiently narrow, but not necessarily to the limit. The area I am taking about can be very naturally partitioned further. I worked in other fields too, and I hope that these papers also contain a couple of new ideas. For sure, they are of a level lower than the one A. Weil had in mind.

On one hand side this personal example shows another extreme way to count the frequency of new ideas. I don’t think that it would be interesting to lower the level further. Many papers and even small lemmas contain some little new ideas (still, much more do not). On the other side, this is important on a personal level. Mathematics is a very difficult profession, and it lost almost all its appeal as a career due to the changes of the universities (at least in the West, especially in the US). It is better to know in advance what kind of internal reward you may get out of it.

As of the timeframe, I think that a new idea is usually understood and used within a year (one has to keep in mind that mathematics is a very slow art) by few followers of the discoverer, often by his or her students or personal friends. Here “few” is something like 2-5 mathematicians. The mathematical community needs about 10 years to digest something new, sometimes it needs much more time. It seems that all this is independent of the level of the contribution. The less fundamental ideas are of interest to fewer people. So they are digested more slowly, despite being easier.

I don’t have much to say about the dynamics (what is the dynamics here?) of progress in mathematics. The past is discussed in many books about history of mathematics; despite I don’t know any which I could recommend without reservations. The only exception is the historical notes at the ends of N. Bourbaki books (they are translated into English and published as a separate book by Springer). A good starting point to read about 20th century is the article by M. Atiyah, “Mathematics in the 20th century”, American Mathematical Monthly, August/September 2001, p. 654 – 666. I will not try to predict the future. If you predict it correctly, nobody will believe you; if not, there is no point. Mathematicians usually try to shape the future by posing problems, but this usually fails even if the problem is solved, because it is solved by tools developed for other purposes. And the future of mathematics is determined by tools. A solution of a really difficult problem often kills an area of research, at least temporarily (for decades minimum).

My predictions for the pure mathematics are rather bleak, but they are based on observing the basic trends in the society, and not on the internal situation in mathematics. There is an internal problem in mathematics pointed out by C. Smorinsky in the 1980ies. The very fast development of mathematics in the preceding decades created many large gaps in the mathematical literature. Some theories lack readable expositions, some theorem are universally accepted but appear to have big gaps in their proofs. C. Smorinsky predicted that mathematicians will turn to expository work and will clear this mess. He also predicted more attention to the history of mathematics. A lot of ideas are hard to understand without knowing why and how they were developed. His predictions did not materialize yet. The expository work is often more difficult than the so-called “original research”, but it is hardly rewarded.


Next post: About some ways to work in mathematics.

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.

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?

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).

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.