Where should I begin?

You are reading out of curiosity, with no plan to work through a course

Start with the essay The Apocalypse According to Galois: the story, with no prerequisites, of what the course proves. The Reading room then lets you leaf through an excerpt of all five volumes, online, without downloading anything.

You are discovering mathematics beyond high school

Start with Volume I, Elements — it begins with number and builds everything else. The essay The Apocalypse According to Galois can come first: it is the story, with no prerequisites, of what the course proves.

You studied once, but high school is far behind

Enter through Volume II: its first part, "Learning to Read Mathematics", re-teaches in six chapters how to read a statement, a definition, a proof. Keep Volume I as a reference for any missing foundations.

You are a mathematics undergraduate

Volume II, Undergraduate, is your working volume — the longest in the collection, the one where proof itself becomes the object of study.

You are preparing a master's degree or the agrégation

Volume III, Graduate, runs from functional analysis to Galois theory and algebraic geometry. The chapters of Volume II reread as refreshers.

You want to understand the Langlands program

Volume IV, Correspondences, is the destination; it assumes from Volume III Galois theory, some commutative algebra, and complex analysis.

You come from physics

Volume V, Mathematical Physics, reads after the functional analysis and Lie groups of Volume III; it is independent of Volume IV.

Assumes : middle-school mathematics · Builds : computation, proof, geometry, complex numbers, functions, probability — the chain ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ

Assumes : Volume I, or solid high-school mathematics · Builds : how to read statements, real analysis, linear algebra, groups, measure theory, first steps into complex and functional analysis

Assumes : Volume II · Builds : functional analysis, distributions, Galois theory, algebraic geometry, manifolds, Lie groups — the structures

Assumes : from Volume III: Galois theory, commutative algebra, complex analysis · Builds : number fields, class field theory, modular forms, Galois representations, up to the Langlands program

Assumes : from Volume III: functional analysis, manifolds, Lie groups — independent of Volume IV · Builds : mechanics, quantum theory, gauge theories, fields, relativity, strings

The algebra and arithmetic track

I (arithmetic, quadratics, complex numbers) → II (polynomials and groups, linear algebra, arithmetic) → III (general algebra, Galois theory, commutative algebra, algebraic geometry) → all of IV.

The analysis track

I (functions, sequences, the integral) → II (real analysis, series, measure and integration, complex analysis) → III (functional analysis, distributions and Fourier, differential equations, dynamical systems) → V for analysis put to work.

The mathematical physics track

I → II (linear algebra, multivariable calculus, probability) → III (functional analysis, manifolds, Lie groups, Riemannian geometry) → V (from symmetries to strings).

Twelve weeks

One volume, thoroughly: all of Volume I (655 pages, about 55 a week), or a chosen part of a later volume.

Twenty-four weeks

A full track to graduate level: your track's chapters across Volumes I–III, about 60 pages a week.

One year

The whole course, 6,200 pages — a serious commitment, about 120 pages a week. Realistic for a sabbatical year or a reading pair.