About the title

About the title

I changed the title of the blog on March 20, 2013 (it used to have the title “Notes of an owl”). This was my immediate reaction to the news the T. Gowers was presenting to the public the works of P. Deligne on the occasion of the award of the Abel prize to Deligne in 2013 (by his own admission, T. Gowers is not qualified to do this).

The issue at hand is not just the lack of qualification; the real issue is that the award to P. Deligne is, unfortunately, the best compensation to the mathematical community for the 2012 award of Abel prize to Szemerédi. I predicted Deligne before the announcement on these grounds alone. I would prefer if the prize to P. Deligne would be awarded out of pure appreciation of his work.

I believe that mathematicians urgently need to stop the growth of Gowers's influence, and, first of all, his initiatives in mathematical publishing. I wrote extensively about the first one; now there is another: to take over the arXiv overlay electronic journals. The same arguments apply.

Now it looks like this title is very good, contrary to my initial opinion. And there is no way back.

Saturday, November 22, 2014

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.

Friday, November 14, 2014

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.

Wednesday, August 13, 2014

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.

Next post: To appear

Tuesday, August 12, 2014

Who will get Fields medals in less than two hours?

Previous post: About expository writing: a reply to posic

At 10:30 p.m. US Eastern Summer time, the winner of this (2014) year Fields medals will be announced in Seoul.

I would like to post my current guess, mostly to have a record of it with the date and time stamp from Google, at least for myself.

As I wrote about one year ago, I believe that I would be able to predict the actual winners if I would know the composition of the Fields medals committee. But I don't. I am not particularly interested in the names of the winners, so I did not attempted to find out the actual winners, who are known for at least three months already, and who are known to the press for at least two weeks already (if the practice of the last two congresses was continued). So, my guess is a guess and not based on any inside sources.

And the winner are (expected to be):

Artur Avila - my confidence is over 95-99%.

One of the winners will be a woman - my confidence is over 95%. This is a pure politics. This deserves a separate discussion. The main obstruction to the Fields medal for a woman is not the discrimination, but the absurd age restriction. Most likely, she is

Sophie Morel - my confidence is over 80%. There are political consideration against her. For example, she would be the 3rd medalist who was a student of Gérard Laumon.

Jacob Lurie - my confidence is about 60%. This is my favorite candidate. He will get it if Harvard has enough political clout now. So, it is a measure of the influence of the Harvard Department of Mathematics, and not of the level of J. Lurie as a mathematician.

Manjul Bhargava - my confidence is less than 50%. If Sophie Morel gets a medal, his chances are much lower than otherwise: two mathematicians from the same university (Princeton).

Following the tradition firmly established since 1990, one of the medals should go a "Russian" mathematician, no matter where she or he is working know and where she or he completed Ph.D. I don't see any suitable candidate. Some people were naming Alexei Borodin, but I was firmly told that he will not get one.

A couple of days ago a strange, apparently unmotivated idea come to my mind: one of the winners will be from Stanford. Some people were naming Maryam Mirzakhani, but, again, a couple of days ago was firmly told that she is not the winner. Her work is interesting and close to my own interests. In my personal opinion, she has some very good results, but nothing of the Fields medal level. I would estimate the number of mathematician of about her level or higher, working in closely related areas, as at least 2-3 dozens. Of course, I am not aware about her most recent unpublished (at least on the web) work.

Next post: And who actually got Fields medals?

Friday, March 7, 2014

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

Thursday, January 2, 2014

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