芝加哥大学本科数学教材推荐(Advanced)

是 否 +2 论坛币

k人 参与回答

经管之家送您一份

应届毕业生专属福利!

求职就业群

赵安豆老师微信：zhaoandou666

经管之家联合CDA

送您一个全额奖学金名额~ !

立即领取

感谢您参与论坛问题回答

经管之家送您两个论坛币！

+2 论坛币

ADVANCEDSpecialized 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)
  

---

FoundationsMac 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 solvingPólya/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 inR^(n+1)?)

General abstract algebraJacobson, 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 representationsAlperin/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.

扫码加我 拉你入群

请注明：姓名-公司-职位

以便审核进群资格，未注明则拒绝

分享0 收藏7 回帖

关键词：Advanced ADVANCE 数学教材推荐 芝加哥大学 VANCE 芝加哥大学 本科

Ring theoryKaplansky, 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 algebraAtiyah/Macdonald, Introduction to commutative algebraMatsumura, Commutative ring theoryEisenbud, Commutative algebra with a view toward algebraic geometryAs 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 theoryWeil, 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).

谢谢分享！！！

Thanks

peter44444xp 发表于 2012-10-11 13:25
谢谢分享！！！

还没完

Combinatorics and discrete mathematicsLovasz, 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 theoryHalmos, 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.

ProbabilityFeller, 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 analysisConway, 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 analysisAndersson, 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 dz = dx + i 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 analysisKatznelson, 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 equationsTaylor, 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 topologyHirsch, 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 topologyMassey, 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 bookGeometry 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 geometrySpivak, 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 theoryMorgan, Geometric measure theory: a beginner's guideMattila, Geometry of sets and measures on Euclidean spacesFederer, 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 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 geometryHarris, 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 1This 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”...

很好！