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.

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“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.

**Undergraduate-level 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.*

**Graduate-level books**

*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.

**Collected papers**

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.