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
Hartshorne is not suitable for learning - it has too many details with no real emphasis. One can become buried alive in Hartshore and never emerge.

ReplyDeleteA much better alternative IMHO is Kempf's very short book. It really contains the essence of algebraic geometry, and since Hartshorne's book (and now the internet) exists, one can usually deal with technicalities when they arise.

Dear Lena Gal,

ReplyDeleteWell, I simultaneously agree and disagree with you. I am not very happy with the Hartshorne’s book, but my reasons are different from yours. For me, the main problem with it is the lack of technicalities, i.e. of complete proofs. Some of the omitted proofs are trivial, some are not. I do not enjoy doing hundreds of exercises; I prefer them to be theorems or lemmas with proofs, allowing me to decide if the statements are obvious, mysterious, etc. and if I want to know proof, to read it, or to reconstruct it myself.

On the other hand, I am very interested exactly in technicalities. More precisely, I am interested more in the new notions themselves than in their applications. This calls for a more elaborated explanation, which I hope to write sometime.

I doubt that such a short book as Kempf's one may contain the essence of algebraic geometry. Algebraic geometry is too broad for this. And I doubt that one can deal with "technicalities" when they arise. If you will get stuck, you will have no clue where in Hartshorne, EGA or SGA look for a lemma. You will suddenly feel that you are in the dark and have no idea what to do next.

Finally, the internet is a communication tool, and it can help you with whatever troubles with Hartshorne you may encounter.