I don’t remember exactly when or why I wrote the list below. I suspect it was a couple of years ago for Part III algebraic geometry students.
If you want to be a mathematician, there’s no substitute for knowing some math. You might as well learn it from great writers.
“To learn to write well, one should read Serre, Bott, Milnor,…” (I think I’m quoting Steve Hurder here, but I believe this too.)
The following list includes both short, readable books that everyone should read and longer reference books.
The standard topics in pure mathematics are: real analysis including Lebesgue integration (I recommend Royden, Real Analysis); complex analysis; topology (I recommend Armstrong, Basic Topology); and algebra including Galois theory. Fourier series are also fundamental; I recommend Dym and McKean, Fourier Series and Integrals, with a variety of applications in a short space.
Part III-level books
Representation theory: Serre, Linear Representations of Finite Groups. Fulton-Harris, Representation Theory (of semisimple Lie algebras, or equivalently of compact Lie groups). By concentrating on examples, Fulton-Harris’s book is wonderfully readable although somewhat long.
Commutative algebra: Atiyah and Macdonald, Introduction to Commutative Algebra. Very clear in a short space.
Number theory: Serre, A Course in Arithmetic. Cassels, Local Fields.
Topology: Bott and Tu’s Differential Forms in Algebraic Topology is a very readable introduction to smooth manifolds and goes far; everyone should read it. Hatcher, Algebraic Topology.
Riemannian geometry: Gallot-Hulin-Lafontaine’s Riemannian Geometry is one of several gentle introductions. Warner’s Foundations of Differentiable Manifolds and Lie Groups is heavier, but is indispensable for giving the only understandable proof of the Hodge theorem for a Riemannian manifold.
Analysis: Royden, Real Analysis. Lieb and Loss, Analysis.
Algebraic geometry: Hartshorne, Algebraic Geometry. Griffiths and Harris, Principles of Algebraic Geometry. These are long references, indispensable for the working algebraic geometer (emphasizing algebraic and analytic approaches, respectively). Huybrechts’s Complex Geometry is a good simplification of Griffiths-Harris.
On more specific topics in algebraic geometry, some outstanding books are Mukai, An Introduction to Invariants and Moduli, and Mumford, Abelian Varieties. There are several other great books (both easier and harder) by Mumford. Borel, Linear Algebraic Groups.
Topology: Milnor’s Characteristic Classes and Morse Theory are magnificent books: short, readable, with a tremendous range of applications. Everyone should read them. There are several other great books by Milnor.
McCleary’s A User’s Guide to Spectral Sequences covers a lot of algebraic topology beyond the basics. Thurston, Three-Dimensional Geometry and Topology.
Symplectic geometry: Arnold, Mathematical Methods in Classical Mechanics.
Homological algebra: Brown, Cohomology of Groups, is an excellent book applying topological ideas to algebra. Weibel, An Introduction to Homological Algebra. Benson, Representations and Cohomology (2 vols.) S. MacLane, Categories for the Working Mathematician.
Number theory: Serre, Local Fields, among several other great books. Lang, Algebraic Number Theory. Miyake, Modular Forms. Silverman, The Arithmetic of Elliptic Curves.
Geometric group theory: Serre, Trees. De la Harpe, Topics in Geometric Group Theory, gives quick treatments of a rich variety of topics.
Analysis: Zimmer, Basic Results in Functional Analysis, treats the fundamental topics and applications in a very short space. Krylov, Lectures on Elliptic and Parabolic Equations in H¨older Spaces, is one of the few graduate-level introductions to serious PDE theory. The big reference books are Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, and Evans, Partial Differential Equations.
Dynamical systems: Walters, An Introduction to Ergodic Theory, is a standard short introduction. Hasselblatt and Katok, An Introduction to the Modern Theory of Dynamical Systems, is the standard big reference book.
Everyone interested in algebraic geometry, number theory, and many aspects of topology and group theory should look at Serre’s Oeuvres: Collected Papers. Atiyah’s Collected Papers are fundamental for topology, with links to analysis and differential geometry.