Post your recommendations/reviews of math books. Which books did you learn easiest from? Which books are wonderful references for the expert, but not a good place to learn for beginners? If you've studied a subject, you've probably got tons of opinions, so post away!
A good companion to this site would be the Chicago undergraduate mathematics bibliography, which covers a wide variety of textbooks up to the level of first- or second-year graduate students.
The standard text for modern algebraic geometry (using schemes) is Hartshorne ISBN 0387902449.
Eisenbud and Harris's "Geometry of Schemes" ISBN 0387986375 is an excellent supplement to chapter 2 of Hartshorne. It's hard (at least for me) to get intuition for schemes just by working through chapter 2 of Hartshorne, but Eisenbud-Harris does a great job of explaining the intuition and motivation behind many of the definitions.
Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea ISBN 0387946802. Though I've only made it through the first few chapters, I've found it be a very readable text with loads of helpful examples and exercises designed to build a hands on feel—very useful in helping me decide if I'll eventually dive into something like Hartshorne.
The same authors have a graduate follow-up called Using Algebraic Geometry ISBN 0387207333.
For a second course, use May's "A Concise Course in Algebraic Topology" ISBN 0226511839, which is also available online. The exposition is elegant, and although the language is fairly elementary, the perspective is quite modern. (May begins by showing that the category of topological spaces is a model category, and then uses this fact systematically to build up homotopy and homology theory.)
I don't know how much it overlaps with the other books mentioned above, but Munkres's Elements of Algebraic Topology ISBN 0201627280 is excellent. Very useful diagrams, concise yet intuition-inspiring language, and structured in a way that revisits key concepts in different contexts. (Note: this is not the same as his other, perhaps more widely known general topology book.)
There is also the book by Bredon ISBN 0387979263, which is good.
MacLane is pretty bad, though widely used. Borceux has three volumes which are very thorough. For higher categories and multicategories (e.g. operads), use Leinster ISBN 0521532159. For infinity-categories (or quasi-categories) use Lurie's Higher Topos Theory (from ArXiv).
David Eisenbud ISBN 0387942696
This is the most commonly used commutative algebra book, at least at Berkeley. Its strength and its weakness is that it tries to be all things to all people. It introduces concepts gently, but also spans a lot of material and contains theorems in pretty much their strongest form. It also emphasizes the relationship with algebraic geometry, both for those who have never seen the subject before and those familiar with the machinery of scheme theory.
If you find Eisenbud too wordy, try Matsumura: ISBN 0521367646.
Atiyah-MacDonald ISBN 0201407515 is a concise and elegant introduction to the basic material. On the downside, it's somewhat dated in its perspective and nowhere near as comprehensive as Eisenbud. It's also expensive, although there seems to be a Spanish translation which is much less expensive: ISBN 8429150080.
Any recommendations/suggestions regarding Lee's Smooth Manifolds ISBN 0387954481 or Spivak's Differential Geometry?
Dummitt and Foote ISBN 0471433349 is a good transition from undergraduate to graduate. It is very thorough, the explanations are good, and it gives a lot of examples.
Weibel ISBN 0521559871 is good for looking up the details of applications, but it doesn't motivate the subject well. If you've never seen homological algebra before, you should read Mirkovic's lecture notes (http://www.math.umass.edu/~mirkovic/A.COURSE.notes/3.HomologicalAlgebra/HA/HA.html), which do a good job of explaining of what's going on, and use more modern language than Weibel. --A.J. 2006-8-24
Rosenberg ISBN 0387942483 is good.
A Shorter Model Theory
Wilfrid Hodges ISBN 0521587131
Every model theory book comes at the subject from a different angle and each of them have their advantages and shortcomings. In terms of learning the subject for the first time, or solidifying your very basic logical foundations for the subject, this book shines. It shocked me at first that they wait until Chapter 5 to introduce the Compactness Theorem, yet I quickly realized the wisdom in this approach. This way, you become familiar with concepts like interpretations and back and forth equivalence, which compare structures. After that, the rest of the book deals with ways of actually constructing structures that have properties you want (like appropriate cardinality, saturation and the like). The book does a fine job keeping these distinction in the concepts clear, as well as covering many syntactical theorems about axiomatizations. However, if you actually want to get your hands more dirty with model constructions and the breakdown of definable sets, you're going to have to look elsewhere.
--Baginski 18:56, 12 August 2006 (PDT)
For an introduction to the real basics in set theory, Enderton's book "Elements of Set Theory" ISBN 0122384407 is very good (and is often used for 135 here at Berkeley). I imagine most graduate students already know all the material in this book, but may have never seen it presented from a set theorist's point of view. You could probably read it in an afternoon.
Moving on from there, the standard introduction to independence proofs in set theory is Kunen's "Set Theory" ISBN 0444868399. It starts off very gently and doesn't assume too much knowledge of other areas of logic. I'd advise readers to skip chapter two (on combinatorics) on a first read and go back and find out about the concepts in there when they're actually needed in the text: this allows you to understand Godel's L construction and the basics of forcing without having to wade through the combinatorics first. An alternative for learning forcing is Cieselski's "Set Theory for the Working Mathematician", which this reviewer feels is a very under-rated book.
After this, the standard choices are Kanamori ISBN 3540003843 and Jech ISBN 3540440852. Jech goes a little slower than Kanamori and has better exercises, but Kanamori is very good for giving you an overview of areas.
I prefer Reed & Simon's Functional Analysis, volume 1 of their series ISBN 0125850506. The approach is rigorous, but not overly general, and they include many interesting examples drawn from quantum mechanics and PDE. It's a good book to dive into the middle of. Rudin's Functional Analysis ISBN 0070542368 is a little too general and the proofs too slick for my taste, but it's beautifully written and very clear, and provides a nice supplement to Reed & Simon.