The content below should roughly correspond to an undergraduate mathematics degree.
How to use this guide
- Do the exercises. They are the main work.
- Reconstruct proofs. If you can't reproduce a proof, you don't understand it yet.
- Use one main book per subject. Alternatives are for different styles or deeper passes.
Part 0: Computational Foundation
Multivariable Calculus
Learn: limits, derivatives, integrals, FTC, integration techniques, sequences/series, Taylor series, vectors, partial derivatives, gradients, multiple integrals, vector fields, line/surface integrals, Green/Stokes/Gauss.
Primary: James Stewart, Calculus, or George B. Thomas, Thomas' Calculus.
Rigorous alternative: Michael Spivak, Calculus.
Linear Algebra (& Computational)
Learn: systems, matrices, determinants, vector spaces, bases, dimension, linear maps, eigenvalues/eigenvectors, diagonalization, inner products.
Primary: Gilbert Strang, Introduction to Linear Algebra + MIT OCW 18.06.
Alternative: Otto Bretscher, Linear Algebra with Applications.
Ordinary Differential Equations
Learn: first order equations, higher order linear equations, systems, series/transform methods, nonlinear systems, stability.
Primary: Boyce & DiPrima, Elementary Differential Equations and Boundary Value Problems.
Qualitative/dynamical alternative: Steven Strogatz, Nonlinear Dynamics and Chaos.
Part 1: Bridge to Proof
Purpose: learn the language of proof before analysis/algebra.
Learn: logic, quantifiers, sets, relations, functions, direct proof, contradiction, contrapositive, induction, cardinality, countability.
Primary: Daniel Velleman, How to Prove It.
Free alternative: Richard Hammack, Book of Proof.
Also good: Kevin Houston, How to Think Like a Mathematician.
Part 2: Core Pure Mathematics
Real Analysis
Build on: calculus, proof bridge.
Learn: real numbers, metric spaces, open/closed/compact sets, sequences/series, continuity, differentiation, Riemann integration, uniform convergence, sequences/series of functions, Stone-Weierstrass, Arzelà-Ascoli.
Primary: Walter Rudin, Principles of Mathematical Analysis.
Gentler first pass: Stephen Abbott, Understanding Analysis.
Alternatives: Charles Pugh, Real Mathematical Analysis, Terence Tao, Analysis I & II.
Abstract Algebra
Build on: proof bridge, computational linear algebra.
Learn: groups, subgroups, cyclic/permutation groups, cosets, Lagrange, homomorphisms, quotient groups, isomorphism theorems, group actions, then rings, fields, polynomials, ideals, Galois theory.
Primary: Michael Artin, Algebra.
Reference: Dummit & Foote, Abstract Algebra.
Gentler (and free): Joseph Gallian, Contemporary Abstract Algebra, Thomas Judson, Abstract Algebra: Theory and Applications.
Point Set Topology
Build on: real analysis, some algebra helpful.
Learn: topological/metric spaces, continuity, connectedness, compactness, quotient spaces, separation axioms, fundamental group, covering spaces.
Primary: James Munkres, Topology.
Linear Algebra (Proof Based)
Build on: computational linear algebra, proof bridge.
Learn: vector spaces and linear maps abstractly, without relying on coordinates.
Primary: Sheldon Axler, Linear Algebra Done Right.
Alternative: Friedberg, Insel & Spence, Linear Algebra.
Honors calculus route: Tom Apostol, Calculus, Volume II.
Part 3: Main Branches
Complex Analysis
Build on: multivariable calculus, real analysis for rigor.
Learn: analytic functions, Cauchy-Riemann, contour integration, Cauchy's theorem/formula, power/Laurent series, residues, conformal maps, argument principle.
Accessible: Brown & Churchill, Complex Variables and Applications.
Rigorous: Stein & Shakarchi, Complex Analysis, Serge Lang, Complex Analysis, Ahlfors, Complex Analysis.
Differential Geometry
Build on: multivariable calculus, linear algebra.
Learn: curves/surfaces, curvature, torsion, first/second fundamental forms, Gauss map, geodesics, Gauss-Bonnet.
Primary: Manfredo do Carmo, Differential Geometry of Curves and Surfaces.
Next: John Lee, Introduction to Smooth Manifolds.
Algebraic Topology
Build on: point set topology, abstract algebra.
Learn: fundamental group, covering spaces, homology, cohomology, Brouwer fixed point theorem, surfaces.
Start: Munkres, Topology Part II.
Then: Allen Hatcher, Algebraic Topology.
Number Theory
Build on: proof bridge, abstract algebra helps.
Learn: divisibility, primes, congruences, Chinese remainder theorem, quadratic reciprocity, arithmetic functions, cryptography basics.
Primary: Niven, Zuckerman & Montgomery, An Introduction to the Theory of Numbers.
Crypto: Hoffstein, Pipher & Silverman, An Introduction to Mathematical Cryptography.
Classic: Hardy & Wright.
Partial Differential Equations
Build on: multivariable calculus, ODEs, analysis helps.
Learn: heat/wave/Laplace equations, separation of variables, Fourier methods, boundary values, Green's functions, characteristics.
Primary: Walter Strauss, Partial Differential Equations: An Introduction.
Fourier Analysis
Build on: real analysis.
Learn: Fourier series/integrals, convergence, Poisson summation, inversion, convolution, uncertainty principle.
Primary: Stein & Shakarchi, Fourier Analysis: An Introduction.
Probability
Build on: calculus for first course, real analysis for measure theoretic version.
Learn: probability spaces, random variables, distributions, generating functions, convergence, LLN, CLT, Markov chains.
First course: Sheldon Ross, A First Course in Probability.
Measure theoretic: Rick Durrett, Probability: Theory and Examples.
Measure Theory
Build on: real analysis.
Learn: Lebesgue measure/integration, convergence theorems, Lp spaces, Radon-Nikodym, intro functional analysis.
Primary: Gerald Folland, Real Analysis, or H. L. Royden, Real Analysis.
Suggested Order
- Multivariable calculus + computational linear algebra.
- Proof bridge with Velleman.
- Real Analysis I + Abstract Algebra I.
- Real Analysis II + Abstract Algebra II.
- Proof based linear algebra + point set topology.
- Branches by interest: complex analysis, differential geometry, number theory, probability, etc.