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Now I'm not a graduate student, but I was hoping to compile a list of what may be considered definitive texts in various branches of mathematics. I'm curious to what books are considered good and useful to people who have a decent level of mathematical maturity (like say a grad student), in hopes that I may understand some of these such books within the next 5-10 years or so. Feel free to add topics that I have forgotten. I'm open to multiple books per subject, too.

I've gone through all questions tagged as "books" and found a few:

  • Algebra:

    • Algebra by S. Lang
    • Algebra by T. W. Hungerford
  • Category Theory:

    • Categories for the Working mathematician by S. Mac Lane
  • Commutative Algebra:

  • Homological Algebra:

  • Representation Theory:

    • Representation Theory: A First Course by W. Fulton / J. Harris
  • Linear Algebra:

  • Real Analysis:

    • Principles of Mathematical Analysis, 3rd. by W. Rudin
    • Real and Complex Analysis by W. Rudin
  • Complex Analysis:

    • Complex Analysis by L. Ahlfors
  • Functional Analysis:

    • Functional Analysis by W. Rudin
  • Measure Theory:

  • General Topology:

    • Topology, 2nd. by J. Munkres
  • Differential Geometry: [Reference]

    • Foundations of Differential Geometry by S. Kobayashi / K. Nomizu (2 vols.)
    • Fundamentals of Differential Geometry by S. Lang
  • Algebraic Geometry:

    • Algebraic Geometry by R. Hartshorne
  • Algebraic Topology:

    • Differential Forms in Algebraic Topology by R. Bott / L. Tu
  • Geometric Topology:

  • Knot Theory:

  • Combinatorics:

  • Graph Theory:

  • Logic:

  • Set Theory:

    • Naive Set Theory by P. Halmos
  • Number Theory:

    • A Classical Introduction to Modern Number Theory by K. Ireland / M. Rosen
  • General / Companion(s):

    • All the Mathematics You Missed: But Need to Know for Graduate School by T. Garrity / L. Pedersen

Thanks.

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18  
I converted this to Wiki. But personally I'm a bit dubious to the utility of this question. For quite a few of the subjects listed (say, at least commutative algebra, functional analysis, measure theory, differential geometry) there are very many good books and people tend to stick with the one they learned from/are familiar with. And exhaustive list probably won't be very enlightening. –  Willie Wong Feb 10 '11 at 12:02
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Agreed. I think this question is much too broad. –  Qiaochu Yuan Feb 10 '11 at 12:51
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You won't ever need to own books on all of those subjects. That's what a library is for. Also, don't go looking for "the one" book on a subject - you'll have no trouble finding 10-15 books on algebraic geometry (for example), all of which have something interesting to say, which is what you want when you're a grad student in that field. –  Gunnar Magnusson Feb 10 '11 at 13:25
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There are different reasons to have a book in your library. For example, Lang's Algebra is an excellent resource, IMHO, but if you want to learn algebra then there are better choices. Are you looking for books that are good resources, or books that you may eventually want to learn from, or a combination thereof? –  Arturo Magidin Feb 10 '11 at 16:21
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I have voted to close: this question is overly broad. Note also that other people have compiled library lists for graduate level mathematics. IMO a much better question would be to ask for links to those preexisting lists. –  Pete L. Clark Feb 10 '11 at 17:06
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16 Answers

There is no such thing as a book which should be in every grad student's library. I know several excellent Ph.D. students who only own a handful of books related to their own field and do just fine. As Gunnar Magnusson pointed out, all the books you listed will anyway be available at the library of any mathematics department.

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Jack Lee, Introduction to Smooth Manifolds.

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Don't you mean by John M. Lee? –  kahen Feb 10 '11 at 16:56
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Eric probably knows him. He goes by Jack, but yes that is the name he publishes under. –  Matt Feb 10 '11 at 17:30
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Overview of Mathematics: The Princeton Companion to Mathematics, editor Timothy Gowers, Princeton U. Press.

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When looking at mathematics textbooks, I always find myself here. I often just read it for the poignant comments. The more difficult textbooks can be found at the bottom. It's nice to see comments about the book from people who have worked through it; amazon.com is notorious for its poor reviews.

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I'm a fan of the Chicago list as well. The main difficulty I have with Amazon reviews of textbooks is that a large number of reviewers there seem to have disliked a class they took using a book, and take it out in their review of the book itself. –  Michael Lugo Feb 10 '11 at 18:08
    
Right! Another favorite of mine: "textbook was in good condition. great seller!" –  Tyler Feb 10 '11 at 19:45
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basic commutative algebra: Atiyah & MacDonald book - Introduction To Commutative Algebra

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linear algebra: Greub - Linear Algebra

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Two wonderful texts on measure theory are :

  • Measure theory by Paul Halmos

  • Measure theory by G.Folland

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Differential Topology: Differential Topology, by Victor Guillemin and Alan Pollack

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Topology:

  1. Ryszard Engelking, General Topology.

  2. Stephen Willard, General Topology.

  3. V. Arhangel'skii, Fundamentals of General Topology.

Commutative Algebra:

  1. R.Y. Sharp, Steps in Commutative Algebra.

Set Theory:

  1. Jean Rubin, Set Theory for the Mathematician.

  2. Michael Potter, Set Theory and its Philosophy.

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Commutative Algebra: Commutative Ring Theory by Matsumura

Algebraic Number Theory: Algebraic Number Theory by Neukirch and Algebraic Number Theory by Cassels and Frohlich, Local Fields by Serre.

Category Theory: Sheaves in Geometry and Logic by Mac Lane and Moerdijk. I know the topic sounds pretty specialized, but I've probably learned more category theory from that book than any other.

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graph theory - Doug West, Introduction to Graph Theory

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I agree with Gunnar Magnusson's comment, repeated in Dan Petersen's answer, that you don't need all these books. But since you asked:

Knots: Rolfsen. Also a black book by someone with a Japanese name, which I forget.

Basic algebraic topology: Hatcher (free online).

Also: Ask the math librarian of any university that has one: he'll be able to tell you which books are popular.

Also: If (for example) you're in algebra and want a good book on topology, ask any algebraist.

Wikipedia serves as a tolerably good resource when you need to look something up which is outside your field but is basic in the field it's in (but doesn't do as well with more specialized things in my experience). Wolfram's MathWorld is also tolerable.

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Homological Algebra - Charles A. Weibel, An Introduction to homological algebra

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Riemannian Geometry: Riemannian Geometry, by Manfredo P. do Carmo.

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Graph Theory - Bela Bollobas, Modern Graph Theory

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Assuming your question the possession of book actually means the understanding of knowledge represented by these book, I think it is a brilliant question, and such thing do exists.

I believe the breadth of knowledge is necessary for a mathematician.

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