Subject Guides

Subject Guide

Mathematics

December 2022

The content below should roughly correspond to an undergraduate mathematics degree.

How to use this guide

  • Do the exercises. Don't read this like a novel!
  • Reconstruct proofs.
  • Use one main book per subject. The stated alternatives are for different styles or deeper passes.

Part 0: Computational Foundation

Multivariable Calculus

Topics: 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)

Topics: 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

Topics: 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

Topics: 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

Topics: 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

Topics: 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.
Free: Joseph Gallian, Contemporary Abstract Algebra, Thomas Judson, Abstract Algebra: Theory and Applications.

Point Set Topology

Topics: topological & metric spaces, continuity, connectedness, compactness, quotient spaces, separation axioms, fundamental group, covering spaces.
Primary: James Munkres, Topology.

Linear Algebra (Proof Based)

Topics: vector spaces and linear maps abstractly, w/o 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

Topics: analytic functions, Cauchy-Riemann, contour integration, Cauchy's theorem, power/Laurent series, residues, conformal maps, argument principle.
Accessible alternative: Brown & Churchill, Complex Variables and Applications.
Rigorous alternative: Stein & Shakarchi, Complex Analysis, Serge Lang, Complex Analysis, Ahlfors, Complex Analysis.

Differential Geometry

Topics: curves & surfaces, curvature, torsion, first & second fundamental forms, Gauss map, geodesics, Gauss-Bonnet.
Read: Manfredo do Carmo, Differential Geometry of Curves and Surfaces.

Algebraic Topology

Topics: fundamental group, covering spaces, homology, cohomology, Brouwer fixed point theorem, surfaces.
Start with: Munkres, Topology Part II.
And then, if you're feeling crazy: Allen Hatcher, Algebraic Topology.

Number Theory

Topics: divisibility, primes, congruences, Chinese remainder theorem, quadratic reciprocity, arithmetic functions, cryptography basics.
Primary option: Niven, Zuckerman & Montgomery, An Introduction to the Theory of Numbers.
Classic alternative: Hardy & Wright.

Partial Differential Equations

Topics: 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

Topics: Fourier series & integrals, convergence, Poisson summation, inversion, convolution, uncertainty principle.
Primary: Stein & Shakarchi, Fourier Analysis: An Introduction.

Probability

Topics: probability spaces, random variables, distributions, generating functions, convergence, LLN, CLT, Markov chains.
Begin with: Sheldon Ross, A First Course in Probability.

Measure Theory

Topics: Lebesgue measure/integration, convergence theorems, Lp spaces, Radon-Nikodym, intro functional analysis.
Primary: Gerald Folland, Real Analysis, or H. L. Royden, Real Analysis.

A Suggested Order

  1. Multivariable calculus, computational linear algebra.
  2. Proof bridge with Velleman.
  3. Real Analysis I, Abstract Algebra I.
  4. Real Analysis II, Abstract Algebra II.
  5. Proof based linear algebra, point set topology.
  6. And then branches by interest, complex analysis, differential geometry, number theory, probability, etc.