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.

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.

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

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).

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

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!

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…

Milnor’s Morse Theory.

Cheeger-Ebin, Comparison Theorems in Riemannian Geometry

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)

Paul Halmos, Lectures on Ergodic Theory

Milnor, Topology from a Differentiable Viewpoint

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

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.

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

Frank Harary: Graph Theory

and

Øystein Ore: The Four-Color Problem

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

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.

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

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

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