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**Chicago undergraduate mathematics bibliography**

Somehow I became the canonical undergraduate source for bibliographical references, so I thought I would leave a list behind before I graduated. I list the books I have found useful in my wanderings through mathematics (in a few cases, those I found especially unuseful), and give short descriptions and comparisons within each category. I hope that this list may serve as a useful “road map” to other undergraduates picking their way through Eckhart Library. In the end, of course, you must explore on your own; but the list may save you a few days wasted reading books at the wrong level or with the wrong emphasis.

The list is biased in two senses. One, it is light on foundations and applied areas, and heavy (especially in the advanced section) on geometry and topology; this is a consequence of my interests. I welcome additions from people interested in other fields. Two, and more seriously, I am an honors-track student and the list reflects that. I don't list any “regular” analysis or algebra texts, for instance, because I really dislike the ones I've seen. If you are a 203 student looking for an alternative to the awful pink book (Marsden/Hoffman), you will find a few here; they are all much clearer, better books, but none are nearly as gentle. I know that banging one's head against a more difficult text is not a realistic option for most students in this position. On the other hand, reading mathematics can't be taught, and it has to be learned sometime. Maybe it's better to get used to frustration as a way of life sooner, rather than later. I don't know.

Reviews not marked with initials, or marked with **[CJ]**, were written by me, Chris Jeris ('98). Other contributors are marked: **[PC]**, Pete Clark ('98); **[PS]**, Pete Storm ('98); **[BB]**, Ben Blander ('98); **[RV]**, Rebecca Virnig ('00); **[BR]**, Ben Recht ('00); **[MG]**, Marci Gambrell ('99); **[YU]**, Yuka Umemoto ('97). Thanks to all of them for their input.

Jump to the elementary, intermediate, or advanced sections.

Warning: Statements about books I haven't looked at in a couple of years may be factually incorrect; please forgive my spotty memory. I don't think I have any really egregious falsehoods in here. I apologize for the appearance of this page; most web browsers have not yet been updated to handle the HTML4 entity set, so fools like me who read the definition write ugly-looking pages.

Enough apologia. Here we go:

**ELEMENTARY**

This includes “high school topics” and first-year calculus.

Contents

- Algebra (4)
- Geometry (2)
- Foundations (1)
- Problem solving (4)
- Calculus (6)
- Bridges to intermediate topics (2)

**Algebra**

** Gelfand/Shen**,

*Algebra*

** Gelfand/Glagoleva/Shnol**,

*Functions and graphs*

** Gelfand/Glagoleva/Kirillov**,

*The method of coordinates*

These three little white books come from the Soviet correspondence school in mathematics, run by I. M. Gelfand for interested people of all ages in the further reaches of the USSR. Rather than trying to be artificially “down-to-earth” in the way Americans do, Gelfand simply assumes that you can understand the mathematics as it's done (and avoids the formal complexities mathematicians are inured to). YSP and SESAME give these out by the carload to their students, who mostly love them. TMoC is notable for its intriguing four-axis scheme for making flat graphs of \(\mathbb R^4\). Overall a fresh, inspiring look at topics we take for granted, and a good thing to recommend to bright younger students or friends (or parents!)

** Cohen**,

*Precalculus with unit circle trigonometry*

**[RV]** I used this book in high school and absolutely loved it. It's very skimpy on proofs, and really should not be used for that sort of insight. However, in terms of understanding how to apply various mathematical concepts it's wonderful. It has a large number of graphs, examples, and easy reference tables. It covers all the algebra, trig, and cartesian geometry that any good high school math sequence should deal with. I have used it for years as a reference book (e.g., what exactly is Cramer's rule again...) Solutions to a number of the problems are in the back, and the problems are not entirely applications.

**Geometry**

** Euclid**,

*The elements*

No, I'm not kidding. At first it's incredibly annoying and tedious to read, but after a while you get into the flow of the language and the style. Euclid teaches you both the power of the modern algebraic methods and the things that are hidden by our instinct to assign a number to a length. Besides, there are wonderful tidbits here and there (did you know that Euclid invented the Dedekind cut?). At least check it out once, to read his proof of the Pythagorean theorem. (Thanks to Jonathan Beere ('95) for convincing me it was worthwhile.)

**[PC]** I have Volume I, and I have to admit I haven't really read it. I do think that I would benefit if someone rammed some of it down my throat though, because nowadays we undergraduates are trained to regard “geometric” as a strong pejorative—the very antithesis of rigor and proof.

** Coxeter**,

*Geometry revisited*

This is a text on “advanced Euclidean geometry”, starting with the numberless classical “centers” of a triangle and proceeding from there. Many good exercises. There are lots of “college geometry” texts you can find this stuff in, but most of them are aimed at math-ed majors; this book and Coxeter's other one (see below) have them all beat.

**[PC]** I like this book. I don't own it but I've flipped through it more than once and I agree that it has a pleasantly non-brain-dead quality to it. There are interesting geometric facts that you probably haven't seen before in here.

**Foundations**

** Rucker**,

*Infinity and the mind*

**[RV]** This is not really a math book. It is a friendly introduction to the concept of infinity, transfinite numbers, and related paradoxes. I'd recommend it to high school students who are intrested in math, but not quite ready to sit down and read though proof after proof of theorems. (In fact, I first read it in high school as part of an independent study math class.) The book does contain some proofs, but not in the rigorous form of a standard math text. It does include more historical background on the concepts than most math texts do, which is nice. Each chapter is accompanied by problems, and an answer key (with explanations) is at the end of the book.

**Problem solving (pre-college)**

*NML problem books*

The MAA publishes a series called “New Mathematical Library” which contains many excellent titles aimed at or below college sophomore level (Geometry revisited is among them). In this series are four books of problems given on the AHSME, one of USAMO and two of IMO problems, all with solutions. We use the AHSME books extensively at YSP; the USAMO and IMO problems still give me a rough time, and are fun if you're looking for frustration one evening.

** Larson**,

*Problem solving through problems*

After you grapple with the IMO problems for a while, turn here to find a book that teaches (as much as any book can) the art of solving them. Cognitive strategies are laid out with examples of problems (mostly from Olympiads and Putnams) to which they apply.

**[PC]** I own this, or at least I did—I haven't seen it since high school. I'm really not a big contest problem-solver, but I did use this book and I think it helped to prepare me for Chicago Mathematics. Lots of good problems, not all of them inane.

** Pólya**,

*How to solve it*

I haven't read this, but it's supposed to be the “classic” version of Larson above.

** Pólya**,

*Mathematics and plausible reasoning, I and II*

**[PC]** These are the “sequels” to Pólya's How to solve it. They are definitely interesting, although their main interest may be psychological/philosophical (only relative to mathematics do philosophy and psychology merge!) I'm not sure that one can really become a significantly better problem-solver by reading a book about the nature of mathematical reasoning, but I admire Pólya for writing an interesting and challenging book about the practice of mathematics; such books are in my opinion too few and far between.

In 1997–98 a few books with the same general theme as Larson, but different problem collections, have been published; I haven't seen any of them.

**Calculus**

Of course, as we all know, the One True Calculus Book is

** Spivak**,

*Calculus*

This is a book everyone should read. If you don't know calculus and have the time, read it and do all the exercises. Parts 1 and 2 are where I finally learned what a limit was, after three years of bad-calculus-book “explanations”. The whole thing is the most coherently envisioned and explained treatment of one-variable calculus I've seen (you can see throughout that Spivak has a vision of what he's trying to teach).

The book has flaws, of course. The exercises get a little monotonous because Spivak has a few tricks he likes to use repeatedly, and perhaps too few of them deal with applications (but you can find that kind of exercise in any book). Also, he sometimes avoids sophistication at the expense of clarity, as in the proofs of Three Hard Theorems in chapter 8 (where a lot of epsilon-pushing takes the place of the words “compact” and “connected”). Nevertheless, this is the best calculus book overall, and I've seen it do a wonderful job of brain rectification on many people.

**[PC]** Yes, it's good, although perhaps more of the affection comes from more advanced students who flip back through it? Most of my exposure to this book comes from tutoring and grading for 161, but I seriously believe that working as many problems as possible (it must be acknowledged that many of them are difficult for first year students, and a few of them are really hard!) is invaluable for developing the mathematical maturity and epsilonic technique that no math major should be without.

Other calculus books worthy of note, and why:

** Spivak**,

*The hitchhiker's guide to calculus*

Just what the title says. I haven't read it, but a lot of 130s students love it.

** Hardy**,

*A course of pure mathematics*

** Courant**,

*Differential and integral calculus*

These two are for “culture”. They are classic treatments of the calculus, from back when a math book was rigorous, period. Hardy focuses more on conceptual elegance and development (beginning by building up R). Courant goes further into applications than is usual (including as much about Fourier analysis as you can do without Lebesgue integration). They're old, and old books are hard to read, but usually worth it. (Remember what Abel said about reading the masters and not the pupils!)

Hardy's book can be freely found here.

** Apostol**,

*Calculus*

This is “the other” modern rigorous calculus text. Reads like an upper-level text: lemma-theorem-proof-corollary. Dry but comprehensive (the second volume includes multivariable calculus).

** Janusz**,

*Calculus*

The worst calculus book ever written. This was the 150s text in 1994–95; it tries to give a Spivak-style rigorous presentation in colorful mainstream-calculus-book format and reading level. Horrible. Take a look at it to see how badly written a mathematics book can be.

**Bridges to intermediate topics**

Springer-Verlag has just begun a new series of texts designed to bring students gently into the realm of abstract mathematics. While there is no shortage of such books, these seem better than average pedagogically; they are all quite talky, include complete solutions to all exercises, and cover sensible (as opposed to traditional) sets of topics. The series is called SUMS, for Springer Undergraduate Mathematics Series. Two so far seem noteworthy: Smith, Introduction to mathematics: algebra and analysis and Johnson, Introduction to logic via numbers and sets. Give them a look.

**INTERMEDIATE**

Roughly, general rather than specialized texts in higher mathematics. I would not hesitate to recommend any book here to honors second-years, but they might not find easy going in some of them. Contents

- Foundations (5)
- General abstract algebra (7)
- Linear algebra (3)
- Number theory (5)
- Combinatorics and discrete mathematics (1)
- Real analysis (10)
- Multivariable calculus (2)
- Complex analysis (5)
- Differential equations (2)
- Point-set topology (5)
- Differential geometry (4)
- Classical geometry (3)

**Foundations**

** Halmos**,

*Naive set theory*

The best book for a first encounter with “real” set theory. Like everything Paul Halmos writes, it's stylistically beautiful. A very skinny book, broken into very short sections, each dealing with a narrow topic and with an exercise or three. It requires just a little sophistication, but no great experience with “real” math; we use this one for YSP kids sometimes too.

** Fraenkel**,

*Abstract set theory*

Fraenkel was the F in ZFC, and he gives a suitably rigorous development of set theory from an axiomatic viewpoint. Unfortunately, for the philosophical foundations of the axioms he refers to another book (Fraenkel and Bar-Hillel, Foundations of set theory), which is missing from Eckhart Library. Good for culture.

** Ebbinghaus/Flum/Thomas**,

*Mathematical logic*

The only logic book I can name off the top of my head, this is the 277 book. I found it readable but boringly syntactic (well, maybe that's elementary logic).

** Enderton**,

*A mathematical introduction to logic*

Look, another logic book! This one might be preferable just because there's much more talking about what's going on and less unmotivated symbol-pushing than in E/F/T. The flip side of that is, the constructions may or may not be epsilon less precise. I'm not a logician; if you are, write some reviews so I can replace these lousy ones!

** Landau**,

*Foundations of analysis*

This is the book that invented the infamous Landau “Satz-Beweis” (theorem-proof) style. There is nothing in this book except the inexorable progression of theorems and proofs, which is perhaps appropriate for a construction of the real numbers from nothing, but makes horrible bathroom reading. Read for culture.

*General abstract algebra*

The situation here is problematic, because there are many good books which are just a little hard to swallow for an average 257 student, but precious few good ones below that. But you learn by doing, so here we go:

\(\color{green}{\text{(Difficulty: moderate)}}\)

** Dummit/Foote**,

*Abstract algebra*

**[PC]** I bought this for 257—I was at the age where I uncritically bought all assigned texts (actually, I may still be at that age; I don't recall passing on buying any course texts recently), but as Chris knows the joke was on me, since we used the instructor's lecture notes and not Dummit/Foote at all. So I didn't really read it that much at the time. I have read it since, since it is one of two general abstract algebra books in my collection. I think it's an excellent undergraduate reference in that it has something to say, and often a lot to say, about precisely everything that an undergraduate would ever run into in an algebra class—and I'm not even exaggerating. I would say this is a good book to have on your shelf if you're an undergraduate because you can look up anything; I used it this fall as a solid supplementary reference for character theory to Alperin and Bell's Groups and representations, and it had an amazing amount of material, all clearly explained. [Warning: there is an incorrect entry in one of the character tables; it's either A*5 or S*5, I can't remember which.] Look elsewhere, particularly below, for a good exposition of modules over a principal ideal domain; D/F's exposition is convoluted and overly lengthy. In fact, overall I would use this book as a reference instead of a primary text, because the idea of reading it through from start to finish scares me. It also has many, many good problems which develop even more topics (e.g., commutative algebra and algebraic geometry).

** Herstein**,

*Topics in algebra*

This is a classic text by one of the masters. Herstein has beautiful and elementary treatments of groups and linear algebra (in the context of module theory). But there is no field theory, and he writes mappings on the right, which annoys many people. Sometimes he suffers from the same flaw of excessive elementarity as Spivak's calculus book, but overall the treatment is quite pretty. Many good exercises. (Not to be confused with Abstract algebra, which is a much-cut version for non-honors classes.)

**[PC]** But this is the book I would use if I were a well-prepared undergraduate wanting to learn abstract algebra for the first time. Wonderful exposition—clean, chatty but not longwinded, informal—and a very efficient coverage of just the most important topics of undergraduate algebra. Think of it as a slimmed down D/F. “No field theory” is certainly an exaggeration; the exposition there is quite brief, and the restriction to fields of characteristic zero obscures the fact that much of the theory presented, including the Galois theory, is the theory of separable field extensions, but even so, this is still the book I open first to remind myself about the Galois theory I'm supposed to know. The last main chapter of the book is quite lengthy and treats linear algebra and canonical forms in detail, which is one of the book's strongest features. Also, there are many supplementary topics—maybe Herstein really doesn't like field theory, since he inserts a section on the transcendence of e early on in his field theory chapter as something of a breather—but there's lots of good stuff to warm the heart of someone who likes to see his algebra applied to actual stuff, especially number-theoretic stuff; the famed Two and Four Squares Theorems are both proved in here!

** Artin**,

*Algebra*

Artin's book is a nontraditional approach to undergraduate algebra, emphasizing concrete computational examples heavily throughout. Accordingly, linear algebra and matrix groups occupy the first part of the book, and the traditional group-ring-field troika comes later. This approach has the advantage of providing many nontrivial examples of the general theories, but you may not want to wait that long to get there. Supposed to be well written, though I haven't read it thoroughly.

\(\color{red}{\text{(Difficulty: higher)}}\)

** Jacobson**,

*Basic algebra I*

Jacobson was my first real algebra book, and I retain an affection for it. The book is very densely written, and his prose has its own beauty but is difficult to get much from at first. The selection of topics is interesting: chapters 1–4 cover groups, rings, modules, fields (modules in the linear-algebra sense, that is, over principal ideal domains), while chapters 5–8 cover extension topics not usually found in general texts. He deliberately avoids modernist abstraction, preferring an explicit construction to a universal property and a commutative diagram (although the universal property is frequently given), and this complicates his notation and prose at times, especially in the module chapter. The field-theory chapter is fantastic. Some of the exercises are deliberately too hard.

** Hungerford**,

*Algebra*

Many people like this book, but I don't. Hungerford covers the standard topics from group, ring, module, and field theory, with a little additional commutative ring theory and the Wedderburn theory of algebras. The field-theory chapter is horrible, and the rest of the book is okay but doesn't excite me. (And the typesetting is bad.)

** Lang**,

*Algebra*

Well, do you like Serge Lang books, or not? Like every other Serge Lang book, this one is uncompromisingly modern, wonderfully comprehensive, and unpleasantly dry and tedious to read. Unlike most other Serge Lang books, this one has exercises, at least.

** Mac Lane/Birkhoff**,

*Algebra*

I keep recommending this book to people because it's the only hard one whose contents correspond well to the 257-8-9 syllabus, and also because I like Mac Lane's treatment of linear and multilinear algebra. Mac Lane and Lang are the only books in this group which treat multilinear (tensor) algebra at all, and believe me, you'll need it eventually. Worth a look to see whether you find Mac Lane's style congenial. Not to be confused with Birkhoff/Mac Lane, A survey of modern algebra (a much shorter and easier book).

**[BR]** I used Mac Lane/Birkhoff's book pretty heavily in Math 257 and 258. Unlike most algebra books I've seen, they don't put all the group theory at the beginning and all of the field theory at the end, but prefer to develop each topic a little bit at a time and then develop it with more depth later. As a result, this book is hard to use as a reference. You can't get past rings without tackling categories and universal constructions which are used heavily throughout the remainder of the text. However, their treatment of categorical algebra is one of the more readable introductions to the theory I've come across.

**Linear algebra**

** Halmos**,

*Finite dimensional vector spaces*

This is a linear algebra book written by a functional analyst, and the crux of the book is a treatment of the spectral theorem for self-adjoint operators in the finite-dimensional case. It's a beautiful, wonderful book, but not a very good reference for traditional linear algebra topics or applications. You also have to read a fair distance before you even see a linear map, and the exercises are mostly too easy, with a few too hard. But this book was where I first learned about tensor products, and why the matrix elements go the way they do and not the other way (Halmos is very careful on this point).

**[PC]** I own this book and read through it often, but it's never taught me linear algebra per se. Let's agree that it's too abstract for a reasonable first introduction to linear algebra; it's really meant for students who already know (some) linear algebra to read through and appreciate one particular, and particularly elegant, presentation of the material. If you want to know about the linear algebra which surrounds functional analysis, then by all means read this book, but much of the material is nonstandard and a bit curious from the perspective of mainstream linear algebra; projections seem to be the most important linear map, and there are many sections lovingly devoted to commuting projections, decomposing projections, etc. I still am not sure why Halmos deifies the [,] as much as he does, and quite honestly, I would learn multilinear algebra anywhere but here.

** Curtis**,

*Abstract linear algebra*

If you can stand terrible typesetting and an unexciting prose style, this tiny little book is a good rigorous reference for traditional linear algebra (i.e., it doesn't assume you're a tree). A nice bonus at the end is the Wedderburn theorem for division algebras over R, although the lack of sophistication makes for some unmotivated technical carpentry. I look in here whenever I can't remember what a positive-definite matrix is.

** Greub**,

*Linear algebra and Multilinear algebra*

You may never need The Book on linear algebra. But one day, you may just have to know fifteen different ways to decompose a linear map into parts with different nice properties. On that day, your choices are Greub and Bourbaki. Greub is easier to carry. End of story.

**Number theory**

** Ireland/Rosen**,

*A classical introduction to modern number theory*

The first half is a coherent, systematic development of elementary number theory, assuming the basics of algebra. In the second half the authors explore more advanced topics of an algebraic/geometric flavor (zeta functions, L-functions, algebraic number fields, elliptic curves). Lots of exercises. This book helped make number theory make sense to me. You will find many introductory number theory texts pitched below I/R, but if you can read I/R, ignore the easy ones.

**[PC]** Yes, this is the standard and to my knowledge the best number theory text that is modern, broad, and reasonably elementary. It's a strange book in that it's really not written at any one level—if you've heard of something called unique factorization, you'll find the first few chapters easygoing material, but the algebraic sophistication rises slowly but surely throughout the book. Eventually you need to be comfortable with rings, fields and Galois theory at the undergraduate level, but they tell you at the beginning of the chapter when they require more background than before. There's an awful lot in here; this was my course text for Math 242 and I used it as one of the texts in a reading class on number theory, and I still haven't read through all the chapters. It's a great example of a book in which the authors have tried and succeeded in bringing advanced material down to the undergraduate level. Some good historical notes, as any self-respecting number theory text should contain. Recommended highly.

** Burn**,

*A pathway into number theory*

**[BB]** The book is composed entirely of exercises leading the reader through all the elementary theorems of number theory. Can be tedious (you get to verify, say, Fermat's little theorem for maybe 5 different sets of numbers) but a good way to really work through the beginnings of the subject on one's own.

** Hardy/Wright**,

*Introduction to number theory*

This is the classic, and Hardy is one of the great expository writers of mathematics. However, I remember that the last time I looked at this book it made no sense to me. If you like number theory you should probably at least look at it.

**[PC]** Oh, here I must fervently disagree (well, okay, maybe it didn't make sense to you at the time, but please go ahead and look again). I say that any student of mathematics should have this book on their shelf. Here's H/W's game: they explain number theory to people who can follow mathematical proofs but have no prior exposure to the subject or any advanced machinery whatsoever—hmm, maybe a little calculus at times, but not always. The one thing they do use is a little asymptotic growth notation, i.e., O, o, and the squiggly line, and for some reason they assume that people will know all about this without much comment. I seem to recall that one chapter towards the beginning is confusing because of this, and when I first bought the book it stymied me (I was sixteen at the time). But it's written so that you don't have to read it in order: they develop just enough theory about almost every branch of (elementary) number theory so that you can see interesting theorems proved. I have jumped around a lot, but over the years I think I've read almost every chapter. I really think it's the #1 “cultural enrichment” book for math students.

** Chandrasekharan**,

*Analytic number theory*

**[PC]** Recommended to me by none other than Professor Narasimhan himself, it's actually a very elementary and readable introduction to the classic theorems of analytic number theory: Chebyshev's Theorem, Bertrand's Postulate, uniform distribution, Dirichlet's Theorem and the Prime Number Theorem. Requires epsilonics and just a little bit of complex function theory.

** Apostol**,

*Introduction to analytic number theory*

**[PC]** If you've been reading this list, you know from Chris that Apostol writes terribly dry books. I've never read anything by him but this one, and it's fine, a bit more elementary than Chandrasekharan and easier to get your hands on (Apostol is a UTM; Chandrasekharan is an out of print Springer international edition). It starts out with a nice introduction to arithmetic functions, including the convolution product, and it covers much the same as the above, only a bit less briskly. A quick route to the proofs of the greatest theorems of 19th century mathematics.

**Combinatorics and discrete mathematics**

** Graham/Knuth/Patashnik**,

*Concrete mathematics*

The first chapter of Knuth's immortal work The art of computer programming is an extensive study of combinatorics and asymptotics. G/K/P is an expanded and friendlier version, which emphasizes teaching the reader to solve things, rather than just showing how they are done. Contains many funny marginal notes from students in the Stanford class which gave birth to the book, as well as tons of great exercises. Not a reference work.

**Real analysis**

\(\color{green}{\text{(Elementary level: metric spaces, continuity, differentiation)}}\)

** Rudin**,

*Principles of mathematical analysis*

The first eight chapters of this little book form the best, cleanest exposition of elementary real analysis I know of, although few UC readers will have much use for the chapter on Riemann-Stieltjes integration. Like Rudin's other books, it is broken into bite-size pieces, so you can prove every statement in the book on your own if you're self-studying. If that isn't enough, there is a large collection of challenging exercises. Some people think Rudin is too skinny and streamlined, but I think it's beautiful. (Ignore chapters 9 and 10, which are a confusing and insufficiently motivated development of multivariable calculus. Chapter 11 is all right for Lebesgue integration, but there are better treatments elsewhere.)

**[PC]** I agree 100% with what Chris says, but I want to add my voice that this is (through chapter 8) the cleanest exposition I have ever seen. I still flip back to this to check things out.

**[BR]** I must insist that Chapters 9 and 10 are not THAT bad. They're worth revisiting if you are tired of Spivak and do Carmo.

** Apostol**,

*Mathematical analysis*

Covers the same material as Rudin, plus a little complex analysis. Apostol assumes (hence, engenders) less maturity on the reader's part, writing most arguments out in “advanced calculus” detail rather than “real analysis” detail, if that makes sense. I find it terribly dry. Nevertheless the book is careful and comprehensive, with many exercises.

** Gelbaum/Olmsted**,

*Counterexamples in analysis*

This little book contains a long list of examples, of strange objects which contradict the things that you think should be true but aren't. It starts off at a very elementary level and gradually builds up to include the Lebesgue theory and R^n. A good thing to have around on your first or second trip through analysis.

\(\color{blue}{\text{(Intermediate level: normed spaces, Lebesgue integration)}}\)

** Kolmogorov/Fomin**,

*Introductory real analysis*

When I started 207 I couldn't see why the material of this book was analysis: here was set theory, some linear algebra, some stuff about normed linear spaces, a little functional analysis... oh, here's that cool integral everyone talks about, but where are the derivatives? Now I know why it's analysis, of course, but the book as a whole is still a perplexing beast to the inexperienced. I think the primary reason it remains a text for 207 is that it costs $13, so why not? The style is distinctively Russian, which puts me off but turns other people on. Extended applications appear occasionally to lend context, but on the whole there is little motivation (and few exercises). The book is also difficult to use as a reference work, because the authors develop only the results they need to get where they're going.

**[PC]** Agreed. But it's cheap and though you may wonder why you're learning so much functional analysis before you see a Lebesgue integral, it's still clear and easy to read, so there's no reason why you shouldn't own it.

** Haaser/Sullivan**,

*Real analysis*

Covers the same material as K/F, with the addition of a chapter relating differentiation to Lebesgue integration (the fundamental theorems of calculus). H/S use the Daniell integral rather than K/F's concrete, bare-hands construction of Lebesgue measure; it's probably good to do it by hand once, but after that forget it. The sequence of topics makes a little more sense than K/F, although the chapter on inner product spaces is lonely at the end, where it lives because they want to do Fourier series. But the book is written in a ho-hum style, and the exercises are too easy. In this H/S shares the flaw of many books at this level, of making too big a deal of a little bit of abstraction which might be new to the reader. I went straight from little Rudin to big Rudin without much of a stop for either of these books.

** Hewitt/Stromberg**,

*Real and abstract analysis*

This is an old, classic book which is worth a look. They develop many concrete classical topics (all those things like Legendre polynomials that you were always curious about) as exercises.

** Dieudonné**,

*Foundations of modern analysis*

This book is a strange bird, the first volume of a nine(!)-volume treatise by one of the original Bourbakistes. I can't really describe it except to say that it's very formalistic, it has many good exercises, it's very hard to relate to other treatments of the subject, and it made a big impression on me.

\(\color{red}{\text{(Graduate level: measure theory, basic functional analysis)}}\)

**Rudin***, *Real and complex analysis*

The first half is the standard reference for real analysis (the second half is reviewed below). It's a very clean treatment of the topics it covers, again in bite-size pieces and with many challenging exercises. Sometimes I get frustrated with the lack of motivation, or with Rudin's habit of proving exactly the lemmas he needs to do something, without any context for the results. Nevertheless it's a good reference or self-study book. Topics: Integration and L^p spaces, Banach and Hilbert spaces, Radon-Nikodym theorem and differentiation, Fubini's theorem, Fourier transforms.

**[PC]** Yes, how wonderful that there's one book whose first half contains all the analysis that you'll ever need to know! This book is advanced and the exposition is austere (“which gives (5). Applying (3) to (4), we get (6)”) but it is absolutely crystalline in its clarity (exception: is its proof of the L^2 inversion theorem for Fourier transforms valid? I'm not so sure.) Isn't this the one math book that every student must buy sooner or later (aside from Hardy and Wright, of course)? Some rainy day you'll discover that the book has a second half and find some very interesting theorems in there, but don't confuse it with a course on complex analysis, because it's a weird-ass treatment of complex analysis viewed through the eyes of a conventional analyst. Think of it as a bonus.

** Lang**,

*Real and functional analysis*

Another Serge Lang book, but a Serge Lang book is about the only place you'll find the inverse function theorem systematically treated for Banach spaces (except Dieudonné, and Lang was a Bourbakiste too). Royden, Real analysis

Royden is like Hungerford for me: a lot of people like it, but it annoys me for a number of semi-silly reasons. He denotes the empty set by 0 (zero) and the zero element of a vector space by lowercase theta. He proves many theorems three times in gradually increasing generality. He leaves whole proofs to the exercises, and then depends on them later in the text. And I don't like his construction of Lebesgue integration. (Nyaah, so there.)

[BR] This is such a terrible book! He leaves the hardest theorems to the reader and proves some really simple-minded things with too much machinery. For example, he assigns the Urysohn lemma for normal spaces as an exercise for the reader and then has to use the Baire category theorem to show that on Banach spaces, linear operators are continuous iff they commute with taking limits. If you have to take 208 or 272, find a supplementary text. You'll be happy you did. Multivariable calculus Spivak, Calculus on manifolds

This is the book everybody gets in differentiation and integration in R^n, and it's a pretty good one, although the integration chapters are hard to read—maybe it was just my first encounter with exterior algebra that made it hard. As usual for Spivak books, clear exposition and lots of nice exercises. Unfortunately this one is old enough to be annoyingly typeset.

**[PC]** I don't really like this book, and I'm a big fan of Spivak in general. Does anybody else think that this rigorous multivariable Riemann integral theory is a dinosaur? And when Spivak starts talking about chains (in chapter four, I think), I don't know what the hell he's talking about. Presumably you could ignore that chapter and use the book as an introduction to differential forms. I can't suggest a substitute at the moment, other than Spivak's Comprehensive introduction volume 1, which is a wonderful book but which I still wouldn't want to read as a first introduction to forms. Come to think of it, I love forms to death, but maybe they're just plain confusing the first time around...
do Carmo, Differential forms and applications

This skinny yellow book has replaced Munkres's Analysis on manifolds as the text for 274, and I'm not sure it's an improvement. It's more like a modernized Calculus on manifolds. I haven't done more than glance through it, but the notation is reputedly horrible, and Spivak is definitely a superior expositor.

**Complex analysis**

** Ahlfors**,

*Complex analysis*

Ahlfors has been the standard text for complex function theory for quite some time. I like it, but he's very classical and concrete in outlook: nary a function space or a norm in the whole book. The exposition is a classic, though.

**[PC]** Everyone lists it; do people actually read it? I'd use Conway instead.
Conway, Functions of one complex variable I

This book starts very, very slow and easy, so if you're rusty on metric spaces or real-variable theory you have no need to worry. Conway's style is to prove things very thoroughly, but relegate the occasional proof to the exercises. The text is more modern than Ahlfors; Conway proves Runge's theorem using Banach space techniques (well, he's an operator theorist). I like the book more for this reason, but I finally sold my copy because the slow pace got to me.

[PC] I like the book, but I hear your criticisms. The chapter on convergence in the compact-open topology, arguably the most important topic in the whole book, is marred by the fact that he mixes metric space theory which is perfectly general with the theory of complex functions. His chapter on Riemann surfaces sort of annoys me too, for the same reason. Maybe just a bit of reorganization would improve this book. But he covers all the theorems that an undergraduate needs to know (and a little more), and he does it without using fancy machinery of any sort: no fundamental groups, no differential forms, no deep theorems from real analysis. [CJ: The Hahn-Banach theorem isn't a deep theorem from real analysis?] Still, I can't help but think that the great American complex analysis book has yet to be written. Narasimhan, Complex analysis in one variable

As we might expect from the famed freshman-eating Narasimhan, this book is much quicker-paced and covers more topics than either of the two above (including a chapter on several variables). Sadly, there are no exercises, but the book is a good reference work. Rudin, Real and complex analysis

Rudin's second half is a treatment of complex analysis even more modern than Conway but even more resolutely non-geometric than Ahlfors. I never really got along with it, for the second reason; also, the selection of topics after the canonical material feels a little random. (Rudin's aim was to bring out the unifying threads in real and complex analysis; thus there is a chapter on Banach algebras near the end.) However, the style is still crystalline, and the exercises are still excellent. Best for confirmed analysts. Palka, An introduction to complex function theory

[YU] The author follows Ahlfors's approach and thus the book is very geometric. After reading this book, I began to like complex function theory. It contains lots of interesting exercises as well as routine ones. Differential equations Arnold, Ordinary differential equations

Yes, Virginia, there is an interesting geometric theory of differential equations (of course!), not just the stuff you see in those engineering texts: stuff about stable and unstable points or manifolds, and other things with a dynamical-systems flavor. Nevertheless there is substantial material on how to reduce a differential equation to linear form and solve it, although no Laplace transform techniques or the like. Arnold explains it all coherently at an advanced-calculus level (manifolds appear at the end), complete with many beautiful diagrams. Another distinctively Russian book—read all the ones I describe that way, and you'll see what I mean. The third edition is substantially different from the second (which I have): the manifolds material is much expanded, and the typesetting is not so nice. Hurewicz, Lectures on ordinary differential equations

A tiny book which covers material similar to Arnold, but more concisely. I haven't read it but it's frequently referenced, and worth a look if you need to know the basic theorems. (If all you need is the basic existence-uniqueness theorem for ODEs, it's also in Spivak volume 1 or Lang, Real and functional analysis.) Point-set topology Munkres, Topology: a first course

Munkres's book is a wonderful first encounter with topology; in fact it begins slowly enough to be a first encounter with abstract mathematics (after a traditional advanced calculus course). Every abstraction is carefully motivated, and there are tons of examples, pictures, and exercises. This is one of those books you could hand to a bright student of any age who knew some calculus (not a bad book to choose if you're coming back to mathematics at age 35). Most of the book is the traditional analysis-topology material, but there is a long last chapter on the fundamental group which covers enough to prove the Jordan curve theorem.

[PC] Yes, Munkres deserves to be the standard undergraduate point-set book. It doesn't have everything, but it has most of the standard topics and it's relentlessly clear. Willard, General topology

But Willard is my topology book of choice. The level of abstraction is deliberately higher, and the book is better organized as a reference than Munkres. It's not nearly as friendly, but it's still clear and well-written (I think an unclear point-set topology book is probably no longer a point-set topology book). Willard is probably the best modern reference for analysis-topology, where “modern” means “excluding Kelley” (see below). You can learn from it too; it's organized bite-size like a Rudin book, so you can prove all but the hard theorems on your own (I did this with an initial segment, and learned a lot). Kelley, General topology

[PS] Let me just say that Kelley's book on topology is horribly old-fashioned—I know because my advisor is forcing me to read it. Half the topics are things which I don't think are as important as they used to be. Nets, filters? I guess they're interesting in and of themselves. On the upside, it does have a nice appendix covering the rudiments of set theory.

[CJ] It is old-fashioned, but it's still the best book on topology for functional analysis, bar none. Nets are surprisingly necessary in infinite-dimensional topological vector spaces! The occasional proof is easier to read once recast in modern language, but doing so is a good learning exercise anyway. And Kelley has the nice habit (emulated less successfully by Willard) of treating substantial pieces of analysis as exercises; two of the exercises to Chapter 2 are titled “Integration theory, junior grade” and “Integration theory, utility grade”. It's really an analysis book disguised as a point-set topology book, but then much of functional analysis is really general topology on spaces that happen to be vector spaces too. Steen/Seebach, Counterexamples in topology

This is a topology ‘anticourse’: a collection of all the screwed-up topological spaces which provide limiting counterexamples to all those point-set topology theorems with complicated hypotheses. It's a classic just for the content, but pretty well written too. This book and Gelbaum/Olmsted (above) are two parts of what should someday be the big book of counterexamples to everything. Read it and see just what you avoid by sticking to differentiable manifolds.

[BR] Steen and Seebach have catalogued 143 of the most disgusting pathological topological creatures. They are invaluable for when you're first learning point set topology and need to understand why the definitions are necessary. They can also come in handy on tests: I used the one-point compactification of an uncountable discrete space three times on my Math 262 final. The text used for 262, Munkres, relies on three counterexamples to disprove everything: the Sorgenfrey line, S_Omega and I x I in the dictionary order. Steen and Seebach let you know that there are tons of other beastly topological spaces which violate the laws of common sense. Dugundji, Topology

[YU]This is a point-set topology book. Less elementary than Munkres, but useful as a reference book for grad students. Differential geometry Guillemin/Pollack, Differential topology

I didn't understand transversality at all until I saw this book. It's a very geometric (as opposed to formalistic), down-to-earth introduction to some of the most mystical areas of smooth manifold theory: transversality and intersection theory. Abstraction is avoided (manifolds are defined as embedded in Euclidean space, which annoys me just a bit), but without hand-waving important distinctions (they are careful to point out that for noncompact manifolds, an injective immersion need not be an embedding, that is, proper too). The last chapter treats integration and Stokes's theorem, but that's not what anyone reads the book for. Beautifully written, and fills an important hole in Spivak volume 1. do Carmo, Differential geometry of curves and surfaces

We used this book for Corlette's differential geometry seminar two years ago (293). I didn't like it all that much because do Carmo is careful to keep the book to a post-advanced-calculus level: everything takes place in R^3, no vector bundles, lots of componentwise calculations. Nevertheless it's a nice treatment of the classical theory of curves and surfaces in space. Read it if you want to know about the Gauss map or the two fundamental forms, but don't want to work all the way through Spivak volume 2. Spivak, A comprehensive introduction to differential geometry, 1

[PC] Volume 1 is the best introduction to smooth manifold theory and differential topology that I know of. Every chapter of this book has come in handy for me at one time or another. Ben and I like to describe the book as “locally readable”: his exposition is very careful, but sometimes he takes too damn long to explain a single concept. Luckily, despite Spivak's efforts to the contrary, you can flip around and read chapter by chapter, and I recommend this. There is so much good stuff in here.

[CJ] Buy it and read it over and over and over. Don't skip the exercises because that's where he puts all the freaky examples. It's true that sometimes he talks too much, but for the loving detail in which he lays out difficult concepts, he can be forgiven. Spivak, A comprehensive introduction to differential geometry, 2

As Spivak puts it at the beginning, “Volume 1 dealt with the ‘differential’ part; in this volume we finally get down to some geometry.” Volume 2 treats the classical theory of curves and surfaces using the modern machinery developed in the first volume, which makes it (for me) a more comfortable read than do Carmo. Spivak is careful to motivate everything historically; surface theory is introduced by a long walk through Gauss's General investigations of curved surfaces (you should really have a copy of it to read this book), and the second half of the book goes through the (convoluted) stages of evolution of the definition of a connection. Not easy reading but every bit as rewarding as Volume 1. Unfortunately there are almost none of the wonderful exercises which characterize the first volume. Classical geometry Coxeter, Introduction to geometry

This is an interesting book which I can't really describe. It contains a number of short treatments of undeniably geometric but nontraditional topics; one fascinating application is the relation between phyllotaxis (the arrangement of plants' leaves around the stem) and generalized Fibonacci-type numbers. Read for culture. Hilbert, Foundations of geometry Hilbert was very interested in finding coherent, minimal axiom systems for parts of mathematics; he was probably inspired by the long debate over Euclid's parallel postulate and the discovery in the late 19th century of consistent non-Euclidean geometries. (The Gödel incompleteness theorems solved negatively one of Hilbert's famous problems.) In this book Hilbert described a correct and complete axiom system for Euclidean geometry, with the dependence relations between axioms exhaustively determined, and then carefully derived most of Euclid from it. It's not a particularly fun read but its existence is philosophically interesting. Hartshorne, [Euclid revisited book]

The algebraic geometer of the famed book from hell (see below) recently finished another modern-Euclid book. I haven't seen it and don't even remember the title, but it might be interesting.

**ADVANCED**
Specialized works, difficulty level unbounded above.

Contents

- Foundations (1)
- Problem solving (1)
- General abstract algebra (1)
- Group theory and representations (5)
- Ring theory (4)
- Commutative and homological algebra (5)
- Number theory (5)
- Combinatorics and discrete mathematics (3)
- Measure theory (2)
- Probability (1)
- Functional analysis (5)
- Complex analysis (6)
- Harmonic analysis (5)
- Differential equations (4)
- Differential topology (3)
- Algebraic topology (7)
- Differential geometry (6)
- Geometric measure theory (4)
- Algebraic geometry (5)

**Foundations**
Mac Lane, Categories for the working mathematician

Pete Clark isn't convinced that the working mathematician needs any category theory at all, but I definitely am! Of course it depends on whether you're interested in something heavily homological, but most people will need at least the basics of adjoints and limits sometime. The book covers substantially more than that, but because examples are drawn from some advanced stuff (rings and Lie algebras appear in the first chapter) you need a fair amount of background to read it. Noteworthy is a section near the end entitled “All concepts are Kan extensions”. Most books on homological algebra will contain a brief summary of category theory, as does Jacobson's Basic algebra II; here you can find it laid out in more detail. Problem solving Pólya/Szegö, Problems and theorems in analysis I and II

These are very old books of very good problems, mostly from analysis, with complete solutions. They're old-fashioned of course, but the polite word is “classical”; worth reading for culture, to prepare for your quals, or (important!) to see if you can still do concrete calculations after four years of brainwashing by abstraction. (Anyone want to compute the n-Hausdorff measure of S^n in R^(n+1)?) General abstract algebra Jacobson, Basic algebra II

This is perhaps the only really advanced general-algebra book; it contains chapters on categories, universal algebra, modules and module categories, classical ring theory, representations of finite groups, homological algebra, commutative algebra, advanced field theory... Readability is uniformly low (unless you really like Jacobson's prose style) and the quality (“sanity”) of the treatments varies; I'd look anywhere else for group representation theory, but as Jacobson is a ring theorist, the structure theory of rings and fields is definitive. (Not the commutative ring stuff though!) I bought it before I really knew whether it was worth having; now I'm not sure, but it's come in handy at surprising times. Of dubious use as a reference, since each chapter is woven rather tightly and he frequently refers to hard results from volume I. Group theory and representations Alperin/Bell, Groups and representations

If you're not into finite groups or their representations, this book contains exactly what you need to know about them. After a quick run-through of what you probably already know, it treats matrix groups (Alperin, like Artin, insists that these are the real examples of finite groups, and I agree), p-groups, composition series, and then basic representation theory via Wedderburn's structure theorem for semisimple algebras. I learned a lot from the matrix-groups chapter. The exposition is nearly as clean and clear as Rudin's, and there are many good exercises (some deliberately too hard, and none marked for difficulty).

[PC] Yep, a solid text for an intro course to group theory (at the graduate level). It's designed so that no more and no less than the entire book gets covered in Math 325, so unlike most math books, I have read this from cover to cover. Rotman, Introduction to the theory of groups

This is a group theorist's group theory book, although it contains no representation theory at all. What I've seen of it looks good (the diagrams on the inside covers are neat, although I have no idea what they mean). But I don't like group theory that much, so I can't say more.

[BR] This was my favorite reference for Murthy's 257 class. Starting with the simplest notions of permutations, Rotman is able to construct everything you ever wanted to know about group theory. If you're just looking for a clear, readable exposition and elegant proofs of the isomorphism theorems or Sylow's theorems, this is a great place to look. And if by some random chance you have need to learn what a wreath product is, you won't need to buy a new book. Gorenstein, Finite groups

[BB] The final word on finite groups prior to 1970. Everything is in here. Very hard reading for a non-specialist, but a good reference for a serious group-theorist. I think Glauberman has it memorized. Humphreys, Introduction to Lie algebras and representation theory

A skinny little book which runs briskly through the basic theorems on Lie algebras and their representations. Note that it says Lie algebras, not Lie groups; there are no smooth manifolds here! There are four copies in Eckhart Library and they're always all checked out, so it must be pretty good; it helps that the alternative works (like Jacobson, Lie algebras) are all very old, thus hard to read. Fulton/Harris, Representation theory: a first course

This is a beautifully concrete introduction to Lie groups and their representations. “First course” in Joe Harris-speak means that the book is driven largely by examination of concrete examples and their characteristics: in fact, the first quarter of the book covers representations of finite groups, as an extended “concrete example” motivating the Lie theory. Nevertheless the book is not easy reading, and you will need a lot of multilinear algebra and some readiness to fill in glossed-over details. But at the end, you will know a lot about why the more advanced general theory behaves as it does. Physicists with a high mathematics tolerance ought to check this one out. Ring theory Kaplansky, Fields and rings

Actually this is three little sheaves (coherent sheaves, even) of lecture notes, bound as a book: one on Galois theory, one on the classical structure theory of (noncommutative) rings, and one on homological dimension theory of rings. Kaplansky's exposition is classic, and for people who (like me) didn't really get Galois theory out of 259, this isn't a bad place to learn it. He has a similar volume called Lie algebras and locally compact groups, which is half structure theory of Lie algebras and half (of all things) a proof that a locally compact topological group has a unique analytic Lie group structure. Anderson/Fuller, Rings and categories of modules

Noncommutative rings have a homological theory very different in flavor from that of commutative rings, namely the structure theory of the categories R-mod and mod-R of left and right modules. I don't really know why I bought this book, because I find the material itself pretty boring. But it's a good exposition, contains category-oriented proofs of most of the classical noncommutative ring theory (as opposed to Lam's book below), and I did use it to give a Math Club talk last year. Morandi, Field and Galois theory

This is an exceedingly gentle but comprehensive course in field theory (a lot more material than the field-theory chapter of a general algebra text). Morandi goes very slowly, and you could probably cover most of the proofs and do them yourself; the beginning exercises are too easy, but there are some good ones too. You might not find the material interesting enough to sustain such length of presentation; if so, look at Kaplansky instead. But it's a good reference if you just need field theory to do something else with (commutative algebra, say). Lam, A first course in noncommutative rings

This is the ring-theory book I should have gotten when I was looking at ring-theory books. Informed by a huge number of examples (many of which I never would have guessed could exist), Lam lays out a beautiful and detailed exposition of the more concrete parts of the theory of noncommutative rings as it exists today. (Some more sophisticated areas, such as the theory of central simple algebras which Jacobson treats in Basic algebra II, are left to a planned second course, now published as Lectures on rings and modules.) Lots of exercises, mostly not too hard. He avoids category-theoretic methods for the most part, which saves the book from turning into the kind of functor catalog that Anderson/Fuller sometimes becomes. Commutative and homological algebra Atiyah/Macdonald, Introduction to commutative algebra Matsumura, Commutative ring theory Eisenbud, Commutative algebra with a view toward algebraic geometry As Pete Clark said, these three are the standard references now, in roughly increasing order of difficulty. Atiyah/Macdonald is short, to the point, and mostly non-homological. Matsumura is the “big Rudin” of commutative algebra: a clear systematic exposition from first principles. Eisenbud is a huge, sprawling monster of a book, which includes almost everything... somewhere. All three have many good exercises, and they complement each other well. Eisenbud is the newest and the most complete reference (and, as a specific objective, includes every result used in Hartshorne's algebraic geometry book), but it can be difficult to wade through so much material to find what you want. Atiyah/Macdonald is probably the best introductory text—or try Kaplansky's book below.

** Kaplansky**,

*Commutative rings*

I list this one separately because it's, well, different. Like Atiyah/Macdonald, this is a small book which takes up commutative algebra from the beginning, largely without homological methods. However, the pace is much brisker, and many results are stated in somewhat idiosyncratic form, since Kaplansky resolutely avoids algebraic-geometric language. He unfortunately refers to the third part of his notes Fields and rings (above) for the homological results he does need.

** Weibel**,

*An introduction to homological algebra*

Without this book I would probably have failed the second half of Kottwitz's Math 327 class. The first half is a systematic exposition of homological algebra, more modern than the standard references: the aim stated is to bring “current technology” in homological algebra to casual users from other disciplines. The second half is devoted to a group of applications, including cohomology of groups (the lifesaver in 327), Lie algebra homology and cohomology, and other stuff. It's reasonably well written and careful in notation (a very important thing in this field). Weibel also takes care not to let too much abstract nonsense go by without an example or three of what in the hell structures he might be talking about.

**Number theory**

** Weil**,

*Basic number theory*

**[PC]** Um, I saw this book in the Coop, was intrigued by the title, and opened it up to a discussion of Haar measure! Not suitable for a first course in number theory, or a second course in number theory, or... It's really hard. Maybe someday I'll get to it.

**[CJ]** It's not that bad, just... brisk. Weil was another of the original Bourbakistes, and his approach to algebraic number theory reflects their devotion to proper foundation: to study global (algebraic number) fields, one must first study local (locally compact) fields, and to study these one begins with topology and measure, etc. I think it's a great book, but it's true you won't learn any number theory you don't already know. You'll discover that you hadn't known what you thought you knew, but now you do.

** Narkiewicz**,

*Introduction to the elementary and analytic theory of algebraic numbers*

This is a huge yellow brick which looks more like a dictionary than a math book. Narkiewicz gives a careful exposition of basic algebraic number theory (in somewhat old-fashioned notation) with more emphasis on the role of (both complex and p-adic) analytic methods than usual. I used it to learn some things about character theory on the p-adics. Notable for its extensive historical notes, unsolved problems lists, and truly immense bibliography.

** Silverman**,

*The arithmetic of elliptic curves*

Silverman's two books (the second is Advanced topics in the arithmetic of elliptic curves) are the standard texts in the subject, and from what I've seen they deserve it. You will need to be thoroughly comfortable with basic algebra and number theory to pick up the first one, however. If you want to learn something about elliptic curves without so much algebraic background, try Koblitz, Introduction to elliptic curves and modular forms (but brush up your complex analysis) or Cassels, Lectures on elliptic curves (and be prepared for a short book that doesn't hold your hand much).

** Koblitz**,

*p-adic numbers, p-adic analysis, and zeta functions*

**[PC]** Interesting, and probably a good place to read up on p-adics.

**[CJ]** I still want to know what a zeta function really is. Koblitz is a good writer, and he'd probably tell me if I read his book...

** Fröhlich/Taylor**,

*Algebraic number theory*

**[PC]** This is the book that I'd love to find time to read from cover to cover. It's advanced in the sense that it's definitely for would-be algebraic number theorists: they cover a lot of ground and basically pride themselves on doing stuff that the other introductory texts don't. For example, they actually talk about cubic, biquadratic and sextic number fields, and complain in their introduction that many number theorists never acquire enough technique to work with anything but quadratic fields. But in terms of prerequisites, it presupposes a solid knowledge of undergraduate algebra, including an acquaintance with modules. I'm biased because I love algebraic number theory, but this book jumped onto my shelf above all the others. There is just so much great stuff in here, and it is written about with enthusiasm and clarity. Only problem is the confusing and oppressive letters that they use for ideals; what's up with that?

**[CJ]** What, the lower-case Fraktur? It's the old standard (grin).

**Combinatorics and discrete mathematics**

** Lovasz**,

*Problems in combinatorics*

**[PS]** You simply must include what Hungarian mathematicians consider the most important math book ever, Laszlo Lovasz's huge tome covering combinatorics from an elementary level to Ph.D. level in one book. It teaches combinatorics the way Hungarians think it should be taught, by doing lots of problems. The problems are very hard, but in the book there are separate sections for problems, hints (which are often quite helpful), and full solutions. Every budding young Hungarian combinatorist spends a year doing every problem in this book sometime before he finishes his Ph.D. As a side treat, the questions are often filled with bits of Hungarian culture, e.g. “How many ways can you pass out k forints to n friends if 1 friend only wants an even number of forints and the rest of them must get at least one?” or “Bela wants to buy flowers for his friend...” Probably the main thing wrong with this book is it's horribly expensive unless you buy it in Hungary, where it's still $60. If you can't find this book in Eckhart, then maybe it's not so important to include it. On the other hand, Babai did help write it, so it is relevant nonetheless.

**[CJ]** A forint is about half a cent these days.

** Stanley**,

*Enumerative combinatorics I*

Combinatorics is maturing from a collection of problems knit together by ad hoc methods (or methods which appear ad hoc to non-combinatorists) into a discipline which is taught and learned systematically. Stanley's book got a rave review in the Bulletin of the AMS as the new standard reference on counting, which really means most of combinatorics; I haven't read it but I've seen it on a whole lot of grad students' shelves. Try it out if G/K/P (above) is too talky for you. The second volume is now out.

** Bollobás**,

*Modern graph theory*

This recent Springer GTM is a substantial revision and expansion of Bollobás's earlier graph theory text. Although I'm not a combinatorist by any stretch of the imagination, it looks like a good book, inviting but not toy.

**Measure theory**

** Halmos**,

*Measure theory*

This was the standard reference for at least two generations of analysts, and it probably still is, because nobody writes books entitled Measure theory any more. Basically it's an abstract analysis text with extra care paid to set-theoretic questions, regularity problems for measures, and a construction of Haar measure. It's a good book, since Paul Halmos wrote it, but it might be considered old-fashioned now. (For a more modern, emphatically measure-theoretic analysis text, check out Bruckner/Bruckner/Thomson, Real analysis.)

** Federer**,

*Geometric measure theory*

Federer's book is listed here because in the last few months, to my great surprise, it has become my reference of choice for basic real analysis (replacing the first half of big Rudin). Chapter 2 (of 5) is entitled “General measure theory”, and it covers chapters 1–3 and 6–8 of big Rudin in the space of eighty pages, together with tons of additional material on group-invariant measures, covering theorems, and all the geometric measures (Hausdorff et al). The presentation is compressed to within epsilon of unreadability, but once you unravel it, it has a powerful elegance. Federer takes great care to give the limits of generality in which each result is true. There are no exercises, but reading the book is hard exercise enough. My one quibble is that even big-name theorems are referenced by number; I would far prefer “by the dominated convergence theorem” to “by 2.3.13” for the rest of the book. If you don't like reading dense books, stay far, far away from Federer, but if you want a complete, powerful reference to measure theory, give it a try.

**Probability**

** Feller**,

*Introduction to probability theory and its applications*

This is the standard text. It splits into two volumes, namely probability before and after it turns into measure theory. What I've read of it is quite well written, and noteworthy for the great care with which it discusses experimental issues (the idea “what sequence of choices corresponds to what mathematical construct” can get sticky when dependence relations are complex). Some of us will need to know some probability someday, and here it is. Alternative references are Shiryaev, Probability (Springer, so cheaper and easier to get, but very Russian) and Billingsley, Probability and measure (by a UC emeritus).

**Functional analysis**

** Conway**,

*A course in functional analysis*

A grad student I knew from 325 saw me leaving the bookstore with this book, and told me it was terrible, that he'd hated it at Dartmouth. I didn't believe him at the time, but now I see what he meant. As in his complex analysis book, Conway develops functional analysis slowly and carefully, without excessive generalization (locally convex spaces are a side topic) and with proofs in great detail, except for the ones he omits. This time around, though, the detail is excruciating (many functional analysis proofs consist of a mass of boring calculation surrounding one main idea) and the notation is simply awful. (The fact that Hilbert spaces are often function spaces is not an excuse to use ‘f’ to denote a general element of a Hilbert space.) The book is not without virtues, but it goes so slowly that I can't see which results are important.

** Dunford/Schwartz**,

*Linear operators*

After all these years, I think Dunford/Schwartz is still the bible of functional analysis; the analysts who did all the exercises in Kelley to learn topology tried to do all the exercises in here, or at least volume 1, to learn about operators. They all failed, although one of the exercises turned into Langlands's doctoral thesis. D/S is too old to be easily read now, but worth looking at for culture.

** Kadison/Ringrose**,

*Fundamentals of the theory of operator algebras*

No, I'm not turning into an operator algebraist (although I might be doing noncommutative geometry some day). The first three-fifths of volume 1 contains a much better treatment of basic functional analysis than I've seen elsewhere, certainly slanted toward operator algebras, but clearly written and interesting (a quality lacking in many functional analysis texts). The book is known for its collection of challenging exercises, which were so popular that K/R wrote up complete solutions to the two volumes and published them as volumes 3 and 4. Unfortunately volume 1 is missing from Eckhart Library.

** Kreyszig**,

*Introductory functional analysis with applications*

Here is a book to look at for a lot of applications and motivation for functional analysis, without a lot of technicalities. I've only looked at it a little bit; it seems to be written more like a physics book, substituting a plausibility argument for an occasional tricky technical proof, but spending a lot of time in explanation. Try it if you have trouble seeing what's really different about the infinite-dimensional case.

** Zimmer**,

*Essential Results of Functional Analysis*

**[BB]** It's a U of C published blue book, and is extremely concise and quickly presents most of the stuff one needs to know. It's certainly not easy—Chapter 0 presents weak derivatives—but it's a good second course.

**Complex analysis**

** Andersson**,

*Topics in complex analysis*

I got through the non-Riemann surfaces part of 314 on this book. It's a skinny Springer Universitext which presents complex analysis at a second-course level, efficiently and clearly, with less talk and fewer commercials. He starts off by defining \(\mathrm dz =\mathrm dx + i \mathrm dy\), which will annoy some people but makes me happy. Later chapters treat more advanced analytic material (Hardy spaces, bounded mean oscillation, and the like). The exercises are pretty tough.

** Gunning/Rossi**,

*Analytic functions of several variables*

This is one of the classic texts on the “real” theory of several complex variables, meaning analytic spaces, coherent sheaves and the whole bit. It's a good book so far as it goes, but there's a lot of hard theory and not a lot of geometric motivation—and no exercises.

** Whitney**,

*Complex analytic varieties*

And this is where you go to learn the “fake” theory of several complex variables, meaning what things actually look like geometrically, with as little machinery as possible. Very concrete. I think there's a law that several-complex-variables books must have no exercises and must use letters as ordinals at some sectioning level.

** Narasimhan**,

*Compact Riemann surfaces*

I put this book here to warn that, although Corlette likes to use it as a 314 text, you should not try to read it until your second or third year of graduate school. It presents the theory of compact Riemann surfaces as someone who already knew the general principles would see it, as a specialization of complex algebraic geometry.

**[PC]** This book lies on my shelf from Math 314, waiting for someone smarter than me to come by and read it. I think I read pages 27 and 28 about 50 times, but that's about it.

** Jost**,

*Compact Riemann surfaces*

If you want to know what Riemann surfaces are and why they're interesting, go here instead. Jost assumes little background; you could probably read this after 207-8-9 with some work.

** Weyl**,

*The concept of a Riemann surface*

Or try this book, which is a beautiful classic but uses terminology and ways of thinking which we consider archaic. Hassler Whitney is credited with the formal definition of a differentiable manifold, and Riemann with the idea (in his Habilitationsschrift; see Spivak volume 2 for a translation), but the first edition of this book was a significant step in its formulation. Read for culture and brain elevation, once you know some substantial complex analysis.

**Harmonic analysis**

** Katznelson**,

*An introduction to harmonic analysis*

And he means analysis... This is a short text on classical harmonic analysis, cheap and pretty readable. There's a rather perfunctory treatment of locally compact groups at the end, but the real emphasis is on the classical theory of Fourier series and integrals, including all kinds of sticky convergence and summation questions.

** Rudin**,

*Fourier analysis on groups*

This is a classic text on commutative harmonic analysis (that is, on locally compact abelian groups). It's a fairly dense research monograph.

** Hewitt/Ross**,

*Abstract harmonic analysis*

H/R is the Dunford/Schwartz of harmonic analysis; this is an immense two-volume set which spends most of a first volume just setting up the generalities on topological groups and integration theory. As such, the recommendation is similar: look at it for culture.

** Stein/Weiss**,

*Introduction to Fourier analysis on Euclidean spaces*

You might think of this as a more advanced Katznelson; it requires a pretty solid comfort with first-year graduate analysis to read.

** Helgason**,

*Groups and geometric analysis*

I found this a fascinating book. At the risk of totally missing the point I might characterize it as the differential-geometric side of noncommutative harmonic analysis (infinite-dimensional representation theory of nonabelian groups). It's about the geometric objects which arise from invariance under symmetries of an ambient space (e.g., the Laplacian is the only isometry-invariant differential operator on the plane). Maybe someday I will actually be able to read it; Helgason's earlier book (below) is a sufficient preparation.

**Differential equations**

** Taylor**,

*Partial differential equations I: basic theory*

I finally learned a little about PDEs, and this book is the first one I'd recommend to any pure mathematicians interested. It's the first volume of a monumental three-volume series covering a wide range of topics in analysis and geometry (yes, Atiyah-Singer is in volume II). Volume I contains the foundational material on Fourier analysis, distributions and Sobolev spaces, application to the classical second-order PDE (Laplace, heat, wave, et cetera), as well as a handy introductory chapter containing all you really need to know about ordinary differential equations! This list of topics doesn't do the book justice, however, since it's packed with interesting little applications and side notes, in the text and the copious exercises. The general consensus among MIT graduate students is that this book, like Federer and Griffiths/Harris, has everything in the world in it.

** Evans**,

*Partial differential equations*

This is a big, fat, talky introduction to PDE for pure mathematicians. It slights some theoretical topics (Fourier transforms and distributions) in favor of an unusually full treatment of nonlinear PDE; the author claims that “we know too much about linear equations and not enough about nonlinear ones,” and his preferences are evident throughout. But it is a good book, written with careful attention to pedagogy and making things make sense to someone new to the field. I like it as a textbook, but Taylor is a better first choice for reference.

** Hörmander**,

*The analysis of linear partial differential operators I*

Here is the book Evans was complaining about; Hörmander's four-volume masterwork contains everything we knew about linear PDE up to the mid-seventies. The first volume is available as a paperback study edition, and makes a good secondary reference on distributions and Fourier transforms. I hope someday to understand the last two chapters, which introduce something called “microlocal analysis” that currently has me fascinated. The book shows little mercy for the reader; distribution theory has some very hard technicalities and Hörmander proceeds pretty briskly. But it's sometimes nice to have a truly definitive reference.

** Olver**,

*Equivalence, invariants and symmetry*

Another book on geometric objects arising from invariance conditions, this one more focused on differential equations. People confused about why the equations of physics look the way they do might try it.

**Differential topology**

** Hirsch**,

*Differential topology*

**[PC]** A solid introduction to differential topology, but maybe a bit bogged down in technical details: a theme of the subject is that arbitrary maps can be approximated by very nice maps under the right conditions. Hirsch has a chapter which he investigates conditions other than “the right ones,” and comes up with some sharpish estimates about when you can approximate what by what. This is sort of interesting, but seems distinctly antithetical to the spirit of “soft” analysis which runs through my veins and the veins of differential topologists everywhere. Why bother? I own the book, and there's some good stuff in it, but in retrospect I'd rather own Guillemin and Pollack, which proceeds a bit more geometrically and far less rigorously. The rigor is optional and can be filled in later.

**[CJ]** I agree with Pete's assessment of the book, but not with his opinions on rigor. Hirsch is a good second differential topology book; after you see how all the touchy-feely stuff goes (move it a little bit to make it transverse), read Hirsch to see how it actually works, and how a nice theoretical framework can be constructed around the soft geometric ideas. I think it's indispensable to see how things are done.

** Lang**,

*Differential and Riemannian manifolds*

Another Serge Lang book, which also contains a proof of the inverse function theorem in Banach spaces (sigh). It's not really human-readable, and I list it mostly because it was the first manifolds book I blundered across in 209. But it has a nice proof of the ODE existence theorem, too.

** Warner**,

*Foundations of differentiable manifolds and Lie groups*

This is a curious selection of material: besides the basic theory of manifolds and differential forms, there is a long chapter on Lie groups, a proof of de Rham's theorem on the equivalence of de Rham cohomology to Cech and topological cohomology theories, and a proof of the Hodge theorem for Riemannian manifolds. It's convenient to have all this stuff here in a single book, but Warner's notation annoys me terribly, and you can find better treatments of any one topic elsewhere.

**Algebraic topology**

** Massey**,

*A basic course in algebraic topology*

Massey wrote two earlier algebraic topology books, Algebraic topology: a first course, and Singular homology theory. This book is their union, minus the last chapter or two of the first book. Thus the first half of the book is a nice, well-grounded treatment of the fundamental group and covering spaces, at a very elementary level (Massey fills in all the material on free groups and free products of groups). The second half is a course on homology theory which is, well, boring. Too slow, too elementary, too talky, and not even very geometric for all that. It'll do, but it's not lovable.

**[PC]** For better or worse, this will probably be your first textbook on algebraic topology. I know Chris doesn't like it very much. The homotopy theory part is fine, but I think the homology/cohomology part could be improved... somehow.

** Fulton**,

*Algebraic topology: a first course*

**[PC]** I own this too, and it's a pleasant book: an algebraic topology book for math students who aren't especially interested in algebraic topology. No, really. I do like algebraic topology, but this book appeals to me too because it takes a holistic and geometric approach to the material; after all, algebraic topology is supposed to be for proving stuff about manifolds and complexes (and other topological spaces of interest, if any), not about chain complexes. There's a lot of interesting stuff here, but because Fulton often contents himself with “the simplest nontrivial case” for fundamental groups, homology, etc., the presentation is less than complete. Great supplementary reading and good treatment of branched covering spaces.

** Bott/Tu**,

*Differential forms in algebraic topology*

This book made algebraic topology make sense to me! Bott/Tu approach cohomology and homotopy theory through the de Rham complex, which means the calculations are all easy to understand and give insight into the geometric situation. The book is not a first course in algebraic topology, as it doesn't cover nearly all the standard topics. What it does cover is beautifully clear, motivated and, well, sensical. They even give a good excuse for spectral sequences, which in my book is a major accomplishment.

** Spanier**,

*Algebraic topology*

Spanier is the maximally unreadable book on algebraic topology. It's bursting with an unbelievable amount of material, all stated in the greatest possible generality and naturality, with the least possible motivation and explanation. But it's awe-inspiring, and every so often forms a useful reference. I'm glad I have it, but most people regret ever opening it.

** Rotman**,

*An introduction to algebraic topology*

**[BR]** You didn't mention this one. I think an appropriate nickname for this one is “Spanier Lite” or maybe “Diet Spanier”, or better still, “Spanier for Dummies.” Rotman was actually a student of the infamous Spanier (and also of Saunders Mac Lane for that matter!). Basically, he stole the table of contents from Spanier's book and tried to write a text that was much less dense and general, but more in depth and more categorical than, say, Massey. I've only read through the first 3 chapters, but anyone who is totally frustrated with having to choose between ultra-elementary and ultra-advanced algebraic topology books should look here.

** Stillwell**,

*Classical topology and combinatorial group theory*

**[PC]** This book is great! No book on this list coincides with my own mathematical esthetics like this one: I checked this book out this summer while I was doing research on surface topology and read it cover to cover: you'll see how geometry relates to topology relates to group theory. I wish this was my first algebraic topology book, because it's full of exciting theorems about surfaces, three-manifolds, knots, simple loops, geodesics—in other words, it's rippling with geometric/topological content intead of commutative diagrams. Let me also recommend Stillwell's book Geometry of surfaces, along the same lines.

** Bredon**,

*Topology and geometry*

Don't be fooled by the word “geometry” in the title; there are two chapters on basic differential topology followed by the best modern course in basic algebraic topology I've seen. Differential geometry and Lie groups supply the occasional example, but there are no metrics to be found! Lots and lots of exercises.

**[PC]** This one gets the Ben Blander seal of approval. From what I've seen, it's an excellent compendium of graduate-level geometry and topology powered by good examples and (again!) actual geometric content.

**Differential geometry**

** Spivak**,

*A comprehensive introduction to differential geometry, 3-5*

The latter three volumes form the ‘Topics’ section of Spivak's masterwork; he treats a succession of more advanced theories within differential geometry, with his customary flair and the occasional stop for generalities. The last chapter is entitled “The generalized Gauss-Bonnet theorem and what it means for mankind”, so that gives you an idea of Spivak's take on geometry. Sadly again, there are no exercises, but the annotated bibliography at the end of volume 5 is immense.

** Helgason**,

*Differential geometry, Lie groups, and symmetric spaces*

The title is a little bit of a misnomer, as this book is really about the differential geometry of Lie groups and symmetric spaces, with an occasional necessary stop for Lie algebra theory. The first chapter is a rapid if rather old-fashioned (no bundles; tensors are modules over the ring of smooth functions) course in basic differential geometry. The rest of the book describes the geometric properties of symmetric spaces (roughly, manifolds with an involutive isometry at each point) in depth. I find the material interesting in itself, and as a lead-in to Helgason's other fascinating book (above). There are many exercises, and solutions at the end!

** Kobayashi/Nomizu**,

*Foundations of differential geometry*

K/N is the standard reference on differential geometry from the sophisticated point of view of frame bundles. The emphasis here is on ‘reference’, unfortunately. I think it's the only book anyone actually uses to look up stuff about principal bundles when they need it, but it's not written as a textbook. The notes and bibliography are very nice, however.

** Rosenberg**,

*The Laplacian on a Riemannian manifold*

**[BB]** A different approach to geometry, through analysis. Lots of exercises integrated critically into the text; proves the Hodge theorem using the heat kernel. Introduces analysis on manifolds. I've only gotten through the first chapter and I've skimmed the rest, so I can't say too much more, but it looks interesting.
do Carmo, Riemannian geometry

**[BB]** A readable and interesting introduction to the subject. It covers some interesting material, such as the sphere theorem and Preissman's theorem about fundamental groups of manifolds of negative curvature, and much more.

** Boothby**,

*Introduction to differentiable manifolds and Riemannian geometry*

I don't know why everyone likes this book so much; maybe because they managed to find it and it contains what they need? It's just another manifolds book, really, and less well-written (lots of annoying coordinates) than most.

**Geometric measure theory**

** Morgan**,

*Geometric measure theory: a beginner's guide*

** Mattila**,

*Geometry of sets and measures on Euclidean spaces*

** Federer**,

*Geometric measure theory*

Okay, so it's a little overkill, but I like geometric measure theory. Here are three books about it, two you should consider reading and one you should consider not reading. Morgan truly is a beginner's guide, and one of the best I've seen to any subject. He introduces the formidable technical apparatus of geometric measure theory bit by bit, leaning on pictures and examples to show what it's for and why we work so hard. Proofs of hard theorems are frequently omitted (mostly referred to Federer). Mattila is a recent book on the theory of rectifiability, and looks good from the little I've seen. Federer is the bible, and it's the densest book I've ever seen, on anything. Everything up to 1969 is in here, and much afterward is anticipated. In addition to the theory of rectifiable sets, Federer develops a powerful homological integration theory, leading to a homology theory for locally Lipschitz sets and maps in \(\mathbb{R}^n\) which is isomorphic on nice sets to the usual homology theories. You can't really learn from it, except that sometimes you have to: the subject is itself very complicated and there are few expositions.

** Falconer**,

*The geometry of fractal sets*

Here is an exposition of the rudiments of geometric measure theory, mostly Hausdorff measures, together with applications to rectifiability and regularity of sets of ugly dimension. A nice little book if you're curious about why it's a cool subject.

**Geometry: algebraic geometry**

** Harris**,

*Algebraic geometry: a first course*

Algebraic geometry is a hard subject to learn, and here is as good a place as any. It has a very different flavor from any other kind of geometry we study in this day and age: lots of results about curves having cusps and intersecting hyperplanes three times. Harris presents a body of classical material (projective varieties over an algebraically closed field of characteristic zero) through analysis of many, many examples, much like his representation theory book. Be warned that much is left out, and you develop your first familiarity with the subject by figuring out what he's really saying. You will also need to be quite comfortable with multilinear algebra. But Harris has a great expository style, and there's a lot of good stuff in all those examples.

** Shafarevich**,

*Basic algebraic geometry 1*

This may be a better place to learn for the first time, as Shafarevich assumes that the language and ways of thought of algebraic geometry are alien to the reader. He proceeds briskly, though, with fewer stops to look around for interesting examples of varieties (ameliorated somewhat by the copious exercises). To make a serious attempt at learning algebraic geometry, you'll probably need both. Shafarevich, like Harris, teaches some of the commutative algebra along the way.

** Mumford**,

*Algebraic geometry I: complex projective varieties*

This book is superficially similar to the previous two (varieties, no schemes) but it's written for mature mathematicians: it's an expository monograph, not a textbook. As such, it's a Good Book in the abstract, but not all that useful to someone looking for guidance. You will need to be solidly comfortable with commutative algebra to begin reading.

** Griffiths/Harris**,

*Principles of algebraic geometry*

A huge, sprawling, beautiful, inspiring, infuriating book. It should be called Principles of analytic geometry, because although the questions are algebraic-geometric, the objects and methods considered are all complex-analytic. This is algebraic geometry over C, the classical case and the one in which existing theory is richest. It's a beautiful and hugely sophisticated theory. G/H treat a vast quantity of it in eight hundred pages, and the treatment is still so compressed that many proofs are quite elliptical. Filling in the gaps requires (or develops) a great deal of maturity. If you're interested in any aspect of algebraic or differential geometry, you should not miss this book—but don't expect any of it to be easy.

** Hartshorne**,

*Algebraic geometry*

Hugh, my algebra TA, described Hartshorne as “the schemes book for the more manly algebraic geometer”. It's the standard exposition of scheme theory, the Grothendieck remaking of algebraic geometry, and it's legendarily difficult, not only the text but the many exercises. The preface to Shafarevich's English edition remarks that “many graduate students (by no means all) can work very hard on Chapters Two and Three of Hartshorne for a year or more, and still know more or less nothing at the end of it.” But, as with most legendarily difficult books, it has its own awesome beauty, and the diligent reader is rewarded. I'm not sure Hartshorne belongs in an undergraduate bibliography, but I did say “difficulty level unbounded above”...

Undergraduate mathematics bibliography, revised 21 January 1999 (142 entries) Christopher Jeris, cjeris@math.mit.edu Back

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