芝加哥大学本科数学教材推荐(Advanced)-经管之家官网！

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)
  

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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.