Comments on mathematics books


Probably no one reads this page so I will keep updating this page if I am ready to comment on the book I read or changed my points of view or if there is a new edition of the book (updated in March 2019).


The updates are due to, I guess, the older I am the more I know about appreciation.


The comments below are personal. They only reflect my personal taste. I have read about 70% of most of the books listed here.


Point-set Topology and Algebraic Topology

Topology by James Munkres

This is one of the most popular textbooks in point-set topology. This book contains many exercises; some of them are not easy. By doing all these exercises correctly you will have a good foundation on topology and proofs.


The author of this book is a famous topologist. However, this book will not excite you very much. His math style is good and plain. Sometimes I think learning topology by reading this book will give you a foundation, but lack of geometric idea (well, point-set topology is not geometric).


Elements of Algebraic Topology by James Munkres

I happen to know that this book is kind of the standard textbook when one takes a graduate course in algebraic topology in US before the book by Hatcher showed up (which is not surprising). Even up to now I think this book is quite concrete. When you need to learn simplicial and singular (co)homology this book is a good place to start. Some other books are either too advanced or have different focuses.


Although this book does not talk about homotopy and does not put too much emphasizes on CW complex, I like the part about simplicial and singular (co)homology.


Algebraic Topology by Allen Hatcher

Everyone puts a high praise on this book, and I think the book deserves it. This book is for beginners with a good background. It is very geometric and really gives you some picture. There are lots of exercises. However, I do not like the part about homotopy. It may be my problem because whenever I see arguments involving finite CW-complex and induction I will yell at the book.


There are some posts in math stackexchange/mathoverflow about `what is the best book in certain subject`. When `certain subject = algebraic topology`, this book always pops up. People who like it say it is very geometric, while for the people who don`t like it they say the proofs are not very rigorous. I agree with both comments, which are not contradictory.


Algebraic Topology: Homology and Homotopy by Robert M. Switzer

I was reading this book and I really like it. This book is quite advanced, but the hard work pays off. It covers homotopy, homology, cohomology, and some K-theory and cobordism, which are hard to find in a single book. Very few exercises, however.


In the past it is one of the very few advanced books in algebraic topology. Nowadays advanced books in algebraic topology are still not many. In the preface of this book the author says after finishing this book the readers are ready for research. I do not know whether this is still an accurate description today but if one wants to do research in algebraic topology, mastering this book is a must.


K-theory by Max Karoubi

There are not many textbooks on topological K-theory. The others I know are Atiyah and Efton Park. Frankly speaking I like this one, though it contains more than I need to know. This book is a little bit dense and has an unusual number of exercises (unusual because K-theory is kind of a research level topic).


It happens that this book contains something I know, and it is written in a very concrete way. I appreciate this book more.



Differential Geometry and related areas

Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo

This is perhaps the best book in differential geometry of curves and surfaces. I use this book as the textbook for my course in undergraduate differential geometry, so you know how much I like it.


One good news about this book is having a Dover edition, which means now this book is affordable, and the pictures in the new edition are better. Sadly do Carmo passed away in 2018 so there will be no newer edition of his books anymore.


A Comprehensive Introduction to Differential Geometry (5 volumes) by Michael Spivak

When I first learn differential geometry (more correctly, smooth manifolds), I did not read this set of books, and I regret. Some people said these books contain too many details and some even said these books make differential geometry harder than it seems to be. I do not agree as differential geometry is not easy and differential geometers always left some details. I always want to finish all these five volumes, but it seems I have no time. These books contain something that is hard to find in other books and, well, I really enjoy reading these books. However, I did not read over 80% of these books. I guess even up to now I just read 8%.


Anyone has done most of the exercises of these books? If you did and happen to read this page, please send me an email. I would like to know the answers of some of the exercises.


More comments: I tried hard to study this book in my spare time but found difficult at some point. Since this book was written during 1960s and had new editions in decades later, I imagine a significant number of differential geometers learn it from these books. Some of these people wrote textbooks on their own, and it is hard not to be influenced by these books. Of course, there will be improvements on the newly written books. For example, the first volume of these books is about the foundation of differentiable manifold. It assumes a good knowledge of point-set topology (although many people feel the other way). I guess the amount of topology assumed by the author is quite large in today`s standard, while it is a piece of cake to the author or any other graduate students in Princeton back in the 60s. Other authors notice this, so the new books on differentiable manifolds avoid topology as much as possible. This gives other new books an advantage over these books by Spivak.


Riemannian Geometry by Manfredo P. do Carmo

I think this book is a good place to learn Riemannian geometry. As is his book on differential geometry of curves and surfaces, you really need to think about what he said rather just read what he said. This makes the readers learn the materials well. This book starts from easy topics and then goes to some not-so-easy topics.


Introduction to Smooth Manifolds by John M. Lee

It is one of the most popular textbooks about smooth manifolds. Popular here means many instructors choose this book as textbook when they teach a course about smooth manifolds. More importantly, it is one of the very few books in geometry whose title really fits to the content (well, there exist many books in geometry with a misleading title). There is a new edition, and it seems to be (which means I have not read in detail) more elaborate and add in some materials (theorems) for the boundary case, which are not easy to find in other books. However, there are some proofs in this book which seem to be too tedious, and I am sorry to say these proofs give the reader a difficult time to see the whole picture.


Now it has a second edition, which is a considerable improvement of the first edition. I love the second edition.


Riemannian Manifolds by John M. Lee

One reason that the second edition of this book is a good one for beginners is the following. First, I am not a Riemannian geometer, so I do not know enough Riemannian geometry. But I need to find a good book on this subject and learn what I need to know. To me most books on Riemannian geometry are either elementary or a bit advanced. I once read a paper not in my field which cites a result that the cut locus has measure zero. I feel upset that I did not know the proof of it, but now Theorem 10.34 of the second edition has a proof.


The Laplacian on a Riemannian Manifold by Steven Rosenberg

This book is an introduction to the analysis on Riemannian manifolds, focusing on its Laplacian and its heat kernel. It also contains a proof of the Gauss-Bonnet-Chern theorem using the heat equation approach, and an introduction to the Atiyah-Singer index theorem. This book contains a number of exercises and is very readable. Well it seems I have promoted this book too much and I should stop here; otherwise there would be a conflict of interest.


Characteristic Classes by John Milnor, James D. Stasheff

People said every book written by Milnor is beautiful and every mathematician, regardless of the area he does, should have this book in his bookshelf. It is very true.


Complex Geometry by Daniel Huybrechts

This is perhaps the most elementary book on complex geometry. It starts with a little bit of several complex variables, and then complex manifolds, holomorphic vector bundle, Kahler geometry, Kodaira vanishing and embedding theorems. It is clearly written and very readable. If you think of Introduction to Smooth Manifolds by John M Lee is the book for beginners in real geometry, then this book is the one for complex geometry.


Differential Geometric Structures by Walter A. Poor

Most books on differential geometry of vector bundles starts with connection, and then curvature and parallel transport and holonomy. This is one of the exceptions which treats parallel transport first and then introduce connection defined in terms of parallel transport. I guess the focus of this book is really on the differential geometry on vector bundles and other structures. Also, you will find some little things in this book that are harder to find in other books.


Spin Geometry by H. Blaine Lawson and Marie-Louise Michelsohn

This book, as its title suggests, has a lot of details on spin geometry. For example, Clifford algebra, Pin group and Spin group, when is a vector bundle spin, etc. Later chapters include the Atiyah-Singer index theorem (essentially the single operator case) and its various applications. Actually, the chapter on the index theorem can be seen as an exposition of the papers by Atiyah-Singer.


Heat Kernels and Dirac Operator by Nicole Berline, Ezra Getzler and Michele Vergne

Previously I said I have mixed feelings about this book, and now I have more positive feeling than negative feeling. This book does not just collect results in local index theory, but it also teaches you the setup of local index theory. More importantly, it contains Bismut`s local family index theorem, and the proofs are given by Nicole Berline and Michele Vergne. I always read this book from time to time (due to my research area), and I always find something that enlightens me, including some techniques I can apply to related areas and something that are missed by other people. I believe that this book is still the only book about local index theory which contains a proof of the local family index theorem up to this date.


I heard some people (actually, just one person, but not me) said whenever you read this book you really want to hit the authors, and this book lacks of motivation. These are in some sense true, but you cannot ask the experts to fill in all the details and no one says it is a textbook. For the motivation I guess understanding the heat equation proof is already a big motivation. If this book does not exist, then probably it is much harder to understand the family index theorem.


When I first learn index theory, I spent n (n>3) years reading the first four and the last two chapters (yes, I was not interested in equivariant things). Then I had some understanding of the family index theorem.


Differential Forms in Algebraic Topology by Loring Tu and Raoul Bott

Everyone says it is a good book, and it`s true. Embarrassingly I only read most of chapter 1 and some of chapter 4. One thing about this book is very concrete and get you to the main point quick.

Riemannian Geometry by Sylvestre Gallot, Dominique Hulin and Jacques Lafontaine

Well, as I said in the comment for `Algebraic Topology` by Allen Hatcher, when you google `what is the best book in Riemannian geometry` this book always pops up. This book is good in the sense that every time a new concept is introduced it is always supported by a bunch of concrete examples, which is a feature that other books in Riemannian geometry may not have, and it contains the solutions of exercises! Moreover, this book covers topics that are hard to find in other books.


It is not hard to see that the style, and perhaps the content of this book is influenced by Marcel Berger. As an off-topic comment, I would love to an English translation of the book `Le spectre d'une variété riemannienne` if it exists.



Real Analysis

Measure and Integral by Richard Wheeden and Antoni Zygmund

For some reason I have taught an undergraduate course in real analysis (which is about Lebesgue measure, integral, etc) at least twice. When I must decide a suitable text for the course, google shows tons of books titled `Real Analysis`. Despite of having different contents, ranging from mathematical analysis to abstract measure theory, it is hard to find a suitable book that fits my taste. Some books are too wordy, and some are too sloppy. This book, however, is a perfect match to me.


Functional Analysis

Introductory Functional Analysis with Applications by Erwin Kreyszig

This book should be one of the easiest and user-friendly books on functional analysis, especially for undergraduates. Although it does not cover topological vector space or Frechet space, it does not require readers to know Lebesgue measure and Lebesgue integral. Moreover, it does have an honest coverage of Banach space, Hilbert space, bounded operators, spectrum and even a little bit of unbounded operators.


I was one of those undergraduates who thinks functional analysis is easy because of picking an easy book. The title of this book already tells you it is an introductory level book on functional analysis (but I guess nowadays no one cares about the title of a book).



An off-topic comment: someone once said a reader owns a great debt to the authors of the books he reads if the reader does not write a book in return. Besides this reason, there are several reasons that I would like to write a book on index theory. The other reasons are mainly due to my learning experience on index theory. Certainly, there are good books on index theory, but one must read several books before reading (or even writing) research papers. Different books have different focuses, coverage, notations and starting points. So, I thought it would be nice if there was a comprehensive introduction to index theory containing all the necessary results on both the topological, geometric and analytical aspects on index theory so that after finishing this hypothetical book one could start doing research without reading the foundational papers. Hopefully this book could come to light before I die.