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.