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
Given that you have previously mentioned R. Stanley (and G.C. Rota) as leading figures in the style of combinatorics you deem valuable to mathematics more broadly, I'm surprised not to see Stanley's Enumerative Combinatorics on this list.

ReplyDeleteDear Shopkins,

DeleteThank you for following my posts.

In the previous post I outlined some guiding principles I used when compiling this list. Stanley's

Enumerative combinatoricsis not on the list because I did not read a sufficiently big chunck of it.I believe that this book is very good and is a must for anybody doing or planning to do the algebraic combinatorics. But, as it should be clear by now, the algebraic combinatorics is not my field. Some problems in my field (as in many many other) are related to combinatorics, and, in particular, to the enumerative combinatorics. Unfortunately, Stanley's book did not helped me. I do not hold this against the book or Stanley:I just would like to say that I would not statisfy even a relaxed form of this guiding principle. Namely, I did not used it in my work.

As a lame excuse I would like to say that I own both volumes, and I am going to order his recent fairly elementary book soon. It was published few months ago by Springer, but only yesterday (I believe) the option to buy a cheap version through our library appeared. It is very likely that I will read some chapters of this one, it looks very attractive.

Correction: "that I would not statisfy even" --> "that it does not satisfy even"

DeleteGreat lists, it include some favorite. I was expecting more books on Algebraic Geometry though :)

ReplyDeleteGreat lists, it include some favorite. I was expecting more books on Algebraic Geometry though :)

ReplyDeleteDear Owl,

ReplyDeleteThank you very much for this list.

You make some interesting comments about calculus and linear algebra. I would be interested to hear your thoughts on the right place of these subjects in mathematics and mathematics education, especially as they are the dominant subjects in American undergraduate programs.

Would you post a list of your favorite undergraduate and even school-level books (for school-level, I mean books such as 'What is Mathematics')?

Also, are there any books you have enjoyed in languages other than English?

ReplyDeleteTo Ravi:

ReplyDeleteThanks! Well, I would be very happy to include more books in algebraic geometry, but I don't know such books. One book is included in the list implicitly under the heading "Other books" by D. Mumford, namely, "The red book of varieties and schemes" (for decades it circulated only as a preprint with the red cover; eventually it was published by Springer with the yellow cover, but with "The red book" being now a part of title). I will add it to the list. There are quite a few presumably good introductory books in algebraic geometry. Most of them are apparently very good if you need to get some idea about what is algebraic geometry, but as textbooks they are to a big extent similar to what is called in the US a "terminal mathematics course": students get some idea of the subject, but are not really prepared to go further and do not have any intent to go further. They may serve as a replacement of Hartshorne's Chapter 1, but you have to read Chapters 2 and 3 anyhow. I did not read any of such books, which disqualifies them from this list. The book (now in two volumes) by Shafarevich is somewhere in between such books and Hartshorne. The book of Ph. Griffiths and J. Harris is almost universally appreciated, but I encountered serious problem when I attempted to use it in teaching. Too many details are missing. Eventually, I encountered a statement included in text without any explanation (not even a theorem or a lemma). It was very plausible, and if not the need to explain it to the students, I would probably go the next phrase without noticing anything. But I had to say something in the class. It wasn't obvious, there was no reference. I found a way to prove it: it immediately follows from a general theory as the codimension 1 case. That theory simplifies a lot in that case, which I did, but remains non-trivial, and it took a month of classes to prove this phrase. Few years later I attempted to use a book by Ph. Griffiths in a course of algebraic curves/Riemann surfaces. I had to abandon it very soon.

The situation with the books in algebraic theory is more suitable for a list "The books I would like to read, but nobody wrote them".

Dear Marius Kempe,

ReplyDeleteThanks for the interest. Your questions about the undergraduate education in US are really important. I would say "really interesting" because they are imposed upon us by external forces. I will try to write something about this issue. My main problem is the need to stop before going into real politics (presidents, congressmen, general trends of the development of the US society). Right now it seems to me that there is no natural point to stop, every statement will need an explanation, and these explanations will quickly progress from mathematics to WWII, and this is not the end.

Anyhow, let me say that I see absolutely no need to teach anything resembling the US calculus courses. Moreover, the majority of students who take mathematics courses will never need it. They hate being forced to take these courses, which is very bad for the public image of mathematics. The standard calculus courses teach students to emulate the most primitive part of computer algebra software. Our university requires students to have calculators if they are taking such a course, but explicitly forbids calculators with the computer algebra capabilities. From my point of view, this is outrageous.

There are good books devoted to the differential and integral calculus. They are at least 50-60 years old, and they are way too difficult for current students (quite naturally, now there are much more students taking calculus than 50 years ago). The Euler's books are very good!

As of undergraduate non-calculus and high-school level books, there are many good ones. Here I can compile only a personal list (more personal than this one); I do not follow such literature systematically.

There is one issue. According to your profile, you are from the UK. What do you mean by "undergraduate"? An undergraduate textbook in UK may be of the graduate level in the US. For example, most of the books in the series "London Mathematical Society Student Texts" will be of the graduate level in the US, but, I believe, of the (top) undergraduate level in UK. Please, correct me if I am wrong. On the other side of the undergraduate education, many books which are at the high school level in Europe, and even were of the high-school level in the US, say, in 1970, are nowadays of the top undergraduate level in the US.

There is a need to have some unambiguous terms for these levels; the internet is a global thing, and Google tell me that people from all over the globe at least visit this blog.

Dear Owl,

ReplyDeleteThanks for your response. I look forward to reading your thoughts.

On the books, I think the average level of the London Mathematical Society Student Texts would make a good upper bound; I like some of those books very much. I don't intend to put any lower bound on the difficulty level.

Correction to my reply to Marius Kempe:

ReplyDeleteI would say "really interesting" --> I would.notsay "really interesting"I like Shafarevich's and Mumfod's books on Algebraic Geometry they are very good, I agree with you.

ReplyDeleteI think W. Rudin's Foundations of mathematical analysis should be Principles of Mathematical Analysis or is it any other book by Rudin.

Dear Ravi,

ReplyDeleteThanks for the correction. Of course, it is "Principles...". Actually, I found this book fairly boring. But it is very efficient. There is one thing in this book which I really don't like: Rudin's ad hoc approach to the differential forms. This part either should be skipped, or not taken seriously and then one will need to learn the standard approach.

Do you plan to write a list "The books I would like to read, but nobody wrote them" from your comment above? It will be very interesting!

ReplyDeletehi sowa... whats new? havent seen you on gowers blog, you arent blocked there are you? you seem to have no comment on his new paper with Ganesalingam .... ripe for criticism dont you think?

ReplyDeletealso there has been a remarkable new computer-based breakthru into the erdos discrepancy problem... any opinion on that? a big writeup of that here, great moments in empirical/experimental math/tcs, breakthru SAT induction idea along with musings on empirical mathematics with lots of refs etc....hey really its not so bad man!!! try it you might like it!!! =)

your blog reminds me at times of a recent quote by aaronson on his blog ripe for parody:

"Most of the time, I’m a crabby, cantankerous ogre, whose only real passion in life is using this blog to shoot down the wrong ideas of others. But alas, try as I might to maintain my reputation as a pure bundle of seething negativity, sometimes events transpire that pierce my crusty exterior. "

Dear vz nuri,

ReplyDeleteYou used another nickname before, vznvzn, right?

Yes, I noticed the paper with Ganesalingam and planned to write about it. The non-technical part of this paper is a very stimulating collection of misconceptions about mathematics. One can write a tract about the nature of mathematics taking this paper as a starting point. I intended to do something more modest, but still more ambitious than refuting them point by point. I planned to write about the background, but this takes time. Unfortunately, I was overwhelmed by more local and more pressing problems. Then I came across some comments by Gowers in Gil Kalai blog, and they look like as a better starting point.

All this is still on the back-burner. And we may expect that the coming announcement of the 2014 Abel prize will provide me with new material, although I hope that it will not.

Being an opponent of Gowers is not my calling. There are other things to do, and other things to blog about, even about mathematics. Perhaps, I should incorporate them into this blog: its popularity, if we trust Google, exceeds my expectations by a very wide margin.

Speaking about the topics of your interest, computer-based stuff, I hardly can offer anything interesting for you. I do not consider such things as a part of mathematics. I will never like them, despite I like a lot of things which are not mathematics. I can offer only a general explanation why this is so.

Dear Marina,

ReplyDeleteSorry for so long delay with a reply. I thought that a reply without that list would be hardly useful, but now it seems that the list is not coming soon.

Yes, I am planning to write such a list. But I have no idea if and when these plans will be realized.