Elementary Calculus
Apostol, Calculus Volume 1
Lax & Terrell, Calculus With Applications;
Courant & John, Introduction to Calculus and Analysis I;
Adams & Essex, Calculus
Multivariable Calculus: Differentiation
Flanigan & Kazdan, Calculus Two
Binmore & Davies, Calculus: Concepts and Methods;
Courant & John, Introduction to Calculus and Analysis II/A;
Duistermaat & Kolk, Multidimensional Real Analysis: Differentiation
Multivariable Calculus: Integration with Forms
Edwards, Advanced Calculus: A Differential Forms Approach
Sjamaar, Manifolds and Differential Forms;
Bachman, A Geometric Approach to Differential Forms;
Duistermaat & Kolk, Multidimensional Real Analysis: Integration
Multivariable Calculus: Both
Hubbard & Hubbard, Vector Calculus, Linear Algebra, and Differential Forms
Fleming, Functions of Several Variables
Shurman, Calculus and Analysis in Euclidean Space
Discrete Mathematics/Intro to Proofs
Rosen, Discrete Mathematics and Its Applications
Epp, Discrete Mathematics with Applications
Introduction to Probability
Feller, An Introduction to Probability Theory and its Applications Vol 1
Blitzstein & Hwang, Introduction to Probability;
Ross, A First Course in Probability
Stirzaker, Elementary Probability
Grinstead & Snell, Introduction to Probability
Combinatorics
Bona, A Walk Through Combinatorics
Bollobas, Combinatorics
Brualdi, Introductory Combinatorics
Polya, Tarjan & Woods, Notes on Introductory Combinatorics
Herbert Wilf, “Generatingfunctionology”
Petersen, T. Kyle, “One, Two, Skip a Few…” (inquiry-based combinatorics)
Van Lint & Wilson, A Course in Combinatorics*;
Introductory Linear Algebra
*Strang, Introduction to Linear Algebra
Solomon, Linear Algebra, Geometry and Transformation;
Beauregard & Fraleigh, Linear Algebra
Dynamical Systems
Palais & Palais, Differential Equations, Mechanics, and Computation
Hirsch, Smale & Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos; Pontryagin, Ordinary Differential Equations;
Hubbard & West, Differential Equations: A Dynamical Systems Approach;
Strogatz, Nonlinear Dynamics and Chaos;
Abraham & Shaw, Dynamics: The Geometry of Behaviour;
Arrowsmith & Place: An Introduction To Dynamical Systems
Introduction to Partial Differential Equations
Logan, Applied Partial Differential Equations;
Borthwick, Introduction to Partial Differential Equations;
Olver, Introduction to Partial Differential Equations
Strauss, Partial Differential Equations: An Introduction*
Abstract Linear Algebra
*Katznelson, A Terse Introduction to Linear Algebra
Valenza, Linear Algebra as an Introduction to Abstract Mathematics
Axler, Linear Algebra Done Right;
Cooperstein, Advanced Linear Algebra;
Weintraub, A Guide to Advanced Linear Algebra
Golan, Foundations of Linear Algebra
Shafarevich & Remizov, Linear Algebra and Geometry
Roman, Advanced Linear Algebra*
Dym, Linear Algebra in Action*
Loehr, Advanced Linear Algebra*
Greub, Linear Algebra*
Introductory Abstract Algebra
*Artin, Algebra
Fraleigh, A First Course in Abstract Algebra;
Goodman, Algebra: Abstract and Concrete
Lauritzen, Concrete Abstract Algebra;
Mac Lane & Birkhoff, Algebra
Gallian, Contemporary Abstract Algebra;
Dummit and Foote, Abstract Algebra
Geometry
Reid & Szendroi, Geometry and Topology
Coxeter, Introduction to Geometry;
Stillwell, The Four Pillars of Geometry;
Guggenheimer, Plane Geometry and Its Groups;
Brannan, Esplen, & Gray, Geometry
Hitchman, Geometry with an Introduction to Cosmic Topology
Fourier Analysis
Stein & Shakarchi, Fourier Analysis
Tolstov, Fourier Series;
Koerner, A First Look at Fourier Analysis
Katznelson, Introduction to Harmonic Analysis
Pereyra & Ward, Harmonic Analysis: From Fourier to Wavelets
Complex Analysis
*Wegert, Visual Complex Functions
Kodaira, Complex Analysis;
Needham, Visual Complex Analysis;
Markushevich, Theory of Functions of a Complex Variable
Introduction to Topology
Brown, Topology and Groupoids;
Janich, Topology;
Sieradski, An Introduction to Topology and Homotopy;
Mendelson, Introduction to Topology;
Croom, Principles of Topology;
Armstrong, Basic Topology;
Crossley, Essential Topology
Flegg, From Geometry to Topology
Munkres, Topology
Classical Differential Geometry
do Carmo, Differential Geometry of Curves and Surfaces;
Bar, Elementary Differential Geometry;
Pressley, Elementary Differential Geometry;
Kuhnel, Differential Geometry;
Guggenheimer, Differential Geometry
Introductory Real Analysis
Rudin, Principles of Mathematical Analysis;
Abbott, Understanding Analysis;
Tao, Analysis Volumes 1 & 2;
Bressoud, A Radical Approach to Real Analysis;
Pugh, Real Mathematical Analysis;
Strichartz, The Way of Analysis;
Krantz, Real Analysis and Foundations;
Berberian, A First Course in Real Analysis;
Ross, Elementary Real Analysis;
Apostol, Mathematical Analysis;
Protter & Morrey, A First Course in Real Analysis
Probability Theory
Billingsley, Probability and Measure
Durrett, Probability: Theory and Examples
Jacod & Protter, Probability Essentials
Chung, A Course in Probability Theory
Introduction to Number Theory
*Ireland & Rosen, A Classical Introduction to Modern Number Theory
Baker, A Concise/Comprehensive Introduction to Number Theory;
Edwards, Higher Arithmetic
Apostol, Introduction to Analytic Number Theory + Modular Functions and Dirichlet Series in Number Theory.
Galois Theory
Stewart, Galois Theory
Mathematical Logic
Kristiansen & Leary, A Friendly Introduction to Mathematical Logic
Enderton, A Mathematical Introduction to Logic;
Johnstone, Notes on Logic and Set Theory
Hamilton, Logic for Mathematicians
Introduction to Set Theory
Enderton, Elements of Set Theory
Devlin, The Joy of Sets
Category Theory
Lawvere & Schanuel, Conceptual Mathematics: A First Introduction to Categories
*Leinster, Basic Category Theory;
Awodey, Category Theory
Riehl, Category Theory in Context
Commutative Algebra
Altman and Kleiman, A Term of Commutative Algebra
Atiyah MacDonald, Introduction to Commutative Algebra
Reid, Commutative Algebra with a View to Algebraic Geometry
Books that Andrew wants everyone to read
Szekeres: A Course in Modern Mathematical Physics;
Escher & Amann: Analysis 1, 2, & 3.
Kurzweil and Stellmacher, The Theory of Finite Groups;
Mac Lane, Categories for the Working Mathematician (1971)
Halmos, Finite-Dimensional Vector Spaces (1942)
Lang, Algebra (1965)
Bourbaki, General Topology, vols I & II (1966)
Chevalley, Theory of Lie Groups (1943)
Warner, Foundations of Differentiable Manifolds and Lie Groups (1971)
Lang, Differential Manifolds (1972)
Rudin, Real and Complex Analysis (1966)
Folland, Real Analysis (1984)
Dunford & Schwartz, Linear Operators, vols I, II & III (1958)
Hewitt & Ross, Abstract Harmonic Analysis, vols I & II (1963)
Yosida, Functional Analysis
Kobayashi & Nomizu, Foundations of Differential Geometry, vols I & II (1963)
Hirsch, Differential Topology (1976)
Spanier, Algebraic Topology (1966)
Gunning & Rossi, Functions of Several Complex Variables (1965)
Hartshorne, Algebraic Geometry (1978)
Shoenfield, Mathematical Logic (1967)
Weil, Basic Number Theory (1967)
Cutland, Computability (1980)
Zeidler, Nonlinear Functional Analysis and its Applications
Taylor, Partial Differential Equations
Kallenberg, Foundations of Modern Probability (1997)
May, A Concise Course in Algebraic Topology (1999)
Lax, Functional Analysis (2002)
*feel free to add grad-level recommendations here*