Math books with longevity

What gives a math book (textbook or otherwise) longevity?   Is there more to a book than just a record of current knowledge, or an aid and reference for a class? Some books continue to be read and reread, and used over and over from one generation to another defying loss of novelty and fashion. What makes a math book great? Is it the subject matter, the presentation, the author’s personal touch or something else that keeps a book relevant over generations?

The answers to these questions may be  personal.  They could be tied up with memories of a favorite spot in the library; the music that was playing as you worked; or the people who were around you. Or perhaps there are universal qualities that make a book great.  Style, elegance, care, quirkiness, beauty, originality…what resonates most with you?

One book that continues to be relevant despite the passage of time is Knots and Links by Dale Rolfsen (AMS Chelsea Series), an introduction to knot theory and low-dimensional topology that was first published in 1976 (Publish or Perish press).   One reason for its lasting significance is that   `Rolfsen’s knot and link table’ is still commonly used to quickly identify knots and links with low crossing numbers.   But what really distinguishes the book, especially for its time, is that it facilitates (in a very effective way) active learning by emphasizing well-chosen hand-drawn illustrations and exercises over long explanations and proofs.


What book was a game-changer for you as a student?  What made/makes it special? Are there any out of print math books you would like to see republished?  Please enter your comments, and mention your favorite book!

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19 Responses to Math books with longevity

  1. Avatar Evelyn says:

    Analytic Functions, the 1960 Princeton University Press book with articles by Nevanlinna, Ahlfors, Bers, and more is an out-of-print book I’d love to see back in print. The Bers article about quasiconformal mappings is probably the basis for everything I actually understand about Teichmüller theory. I don’t know if it’s really great or if it’s just because I spent so much time with it as a graduate student, but I love that article.

  2. Avatar Kevin Knudson says:

    Ken Brown’s Cohomology of Groups and Griffiths and Morgan’s Rational Homotopy Theory. Grad school wouldn’t have been the same without them.

  3. Avatar Barbara says:

    Ha! I get to be the first to mention Hartshorne’s Algebraic geometry (1977) and of course the whole Grothendieck (and school) EGA (1960-67) and SGA series (1960-1977)!
    My very favorite, both because of the very high quality of the exposition and its continuing usefulness along the decades, is Fulton’s Intersection Theory (1984).

  4. Avatar Dick Palais says:

    Michael Spivak’s Comprehensive Introduction to Differential Geometry, Vol. 1

  5. Avatar Matilde Marcolli says:

    The first serious math book I read were the three volumes of Dubrovin, Novikov and Fomenko, “Modern Geometry” … couldn’t put it down, a real page-turner!

  6. Avatar Alec Kercheval says:

    Kaplansky’s Set Theory and Metric Spaces. I read that in high school and it sparked a long interest in logic/set theory — I still use the book for Intro to proof courses.

    Another one: Milnor’s Topology from a differentiable viewpoint. That sparked 15 years worth of my career as a mathematician…

  7. Avatar Deane says:

    Milnor’s Morse Theory.

  8. Avatar Deane says:

    Cheeger-Ebin, Comparison Theorems in Riemannian Geometry

  9. Avatar Dan Abramovich says:

    I want to endorse the idea of the short, self contained book which is perfect for one course. Essential are clear exposition and good exercises. This is what a messy lecturer like me needs. Here are a few examples:

    Michael Atiyah and Ian Macdonald: Introduction to Commutative Algebra (1969)

    Pierre Samuel: Algebraic Theory of Numbers (1970)

    William Fulton: Algebraic Curves: An Introduction to Algebraic Geometry (1969)

    Daniel Marcus: Number Fields (1977)

  10. Avatar Jayadev Athreya says:

    Paul Halmos, Lectures on Ergodic Theory

    Milnor, Topology from a Differentiable Viewpoint

    • Avatar Jayadev Athreya says:

      oh, and Mark Kac, Statistical Independence in Probability, Analysis, and Number Theory .

  11. Avatar Hossein Hosseini Giv says:

    As an analyst I want to mention Walter Rudin’s “Principles of Mathematical Analysis”. The book owes its popularity and longevity to its precise and concise presentation. It gives the reader a lot of information, and it is the reader (or perhaps the instructor) who is responsible for finding the necessary insight. This approach works well for talented students.

  12. Avatar Andy says:

    A few of my favorites that have not been mentioned yet:

    Thurston, Three-dimensional geometry and topology

    Hubbard, Teichmuller theory vol I

    Reed & Simon, Functional analysis

    Miranda, Algebraic curves and Riemann surfaces

    Narasimhan, Compact Riemann surfaces

    Milnor & Stasheff, Characteristic classes

    James, The representation theory of the symmetric group

  13. Avatar Roald says:

    Frank Harary: Graph Theory
    Øystein Ore: The Four-Color Problem

  14. Avatar Mason Porter says:

    The monograph by Guckenheimer & Holmes and the dynamical systems textbook by Steve Strogatz continue to be timely introductions (at different levels) to dynamical systems.

  15. Avatar Eduardo L says:

    Spivak’s “Calculus on Manifolds” is a masterpiece. It is not as detailed as his Comprehensive Introduction to Differential Geometry, but it succeeds in explaining the basic and fundamental topics in a delightful exposition.

  16. Avatar Eder says:

    Probability, Random Variables, and Stochastic Processes by Papoulis, Athanasios 1965.

  17. Avatar Chester says:

    Advanced Ordinary Differential Equations by A. G. Kartsatos, 1980.

  18. Avatar Tim Kohl says:

    I have a few in this category, in no particular order:
    Munkres – Topology [the red book]
    Fraleigh – A First Course in Abstract Algebra
    Rotman – Group Theory
    Weilandt – Finite Permutation Groups
    Burton – Elementary Number Theory
    Hilton&Stammbach – A Course in Homological Algebra
    Curtis&Reiner – Representation Theory of Finite Groups and Associative Algebras
    Waterhouse – Introduction to Affine Group Schemes
    Sweedler – Hopf Algebras
    Samuel – Algebraic Theory of Numbers

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