What should be in every grad student's library?

Question

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.

Answer 1

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.

Answer 2

Overview of Mathematics: The Princeton Companion to Mathematics, editor Timothy Gowers, Princeton U. Press.

Answer 3

Jack Lee, Introduction to Smooth Manifolds.

Answer 4

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.

Answer 5

basic commutative algebra: Atiyah & MacDonald book - Introduction To Commutative Algebra

Answer 6

linear algebra: Greub - Linear Algebra

Answer 7

Two wonderful texts on measure theory are :

• Measure theory by Paul Halmos

• Measure theory by G.Folland

Answer 8

graph theory - Doug West, Introduction to Graph Theory

Answer 9

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.

Answer 10

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

Answer 11

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

Answer 12

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.

Answer 13

Graph Theory - Bela Bollobas, Modern Graph Theory

Answer 14

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