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.

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.

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.

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.

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.

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.

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.

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

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.

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

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.

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.

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.

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.

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.

[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.

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.

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.

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.

[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.

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 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.

[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.

[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.

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.

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.

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!

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.

[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.

[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.

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.

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.

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.

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.

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.

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.

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”...