Graduate level textbooks: A list - the second part

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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.

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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.


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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