Mathematics Textbooks for Self Study -- A Guide for the Autodidact
This is a list of suggested textbooks that a student can use to learn about a
topic on their own, or to supplement the text used in a class.
This list is not meant to be comprehensive. It is designed to contain recommendations
for the standard subjects in U.S. mathematics education that are covered during the
undergraduate years and the first year or two of graduate study. In particular, the
list does not make recommendations for advanced topics (such as C*-algebra theory
or Homological Algebra) or specific topics (such as Knot theory or Coding theory) that
are not covered in a majority of programs, but might be found in topics courses.
This list is also meant to recommend a few of the best books for learning
a topic, not to make an exhaustive to record of all the introductory texts on that topic.
(However, if you think something missing from the list should be there,
just let me know.)
Under each topic I have listed some of the books I've found most useful, as well as
others that are likely to be on the recommendation lists of most mathematicians.
KEY
Books in Blue = Highly recommended (by me) as a first exposure to the topic.
Books in Green = Elementary, accessible with little background.
Books in Red = Difficult to read, but considered a standard and worthwhile
if you can follow it.
Books in Purple = Older and considered a classic; may be difficult
to read due to antiquated notation or terminology, but also contains useful
material not found in newer books.
Books labeled (DOVER) are published by Dover Publications, also known as Dover Books,
which primarily publishes reissues; i.e., books no longer published
by their original publishers that are often, but not always, in the
public domain. Dover books are very inexpensive, often in the range of $10--$20.
Abstract Algebra
Undergraduate
- Contemporary Abstract Algebra by Joe Gallian
- A First Course in Abstract Algebra by Joseph J. Rotman
- Abstract Algebra by I.N. Herstein
Graduate
- Abstract Algebra by David S. Dummit and Richard M. Foote
- Algebra by Thomas W. Hungerford
- Topics in Algebra by I.N. Herstein
- Algebra by Serge Lang
- Algebra by Michael Artin
- Advanced Modern Algebra by Joseph J. Rotman
- Basic Algebra I, Basic Algebra II, and Basic Algebra III by Nathan Jacobson (DOVER)
- Field and Galois Theory by Patrick Morandi
Real Analysis
Undergraduate
- Elementary Analysis: The Theory of Calculus by Kenneth Ross
- Principles of Mathematical Analysis by Walter Rudin
- Understanding Analysis by Stephen Abbott
- Real Mathematical Analysis by Charles Chapman Pugh
- Metric Spaces by E.T. Copson
- Real Analysis by N.L Carothers (DOVER)
Graduate
- Real and Complex Analysis by Walter Rudin
- Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland
- Real Analysis by Richard F. Bass
- The Elements of Integration and Lebesgue Measure by Robert G. Bartle
- Measure Theory by Donald L. Cohn
- Measure Theory by J.L. Doob (Particularly good for students interested in Probability Theory)
- Measure Theory by Paul R. Halmos
- Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
by Elias M. Stein
- Princeton Lectures in Analysis by Elias M. Stein and Rami Shakarchi
Book 1: Fourier Analysis: An Introduction
Book 2: Complex Analysis
Book 3: Real Analysis: Measure Theory, Integration, and Hilbert Spaces
Book 4: Functional Analysis: Introduction to Further Topics in Analysis
A series of four textbooks that aims to present, in an integrated manner,
the core areas of analysis.
Complex Analysis
Undergraduate
- Complex Variables and Applications by James Brown and Ruel Churchill
- A First Course in Complex Analysis With Applications by Dennis Zill and Patrick Shanahan
- Visual Complex Analysis by Tristan Needham
Graduate
- Real and Complex Analysis by Walter Rudin
- Functions of One Complex Variable I, II by John B. Conway
- Function Theory of One Complex Variable by Robert E. Greene and Steven G. Krantz
- Function Theory of Several Complex Variables by Steven G. Krantz
- Complex Analysis by Lars Ahlfors
- Princeton Lectures in Analysis by Elias M. Stein and Rami Shakarchi
Book 1: Fourier Analysis: An Introduction
Book 2: Complex Analysis
Book 3: Real Analysis: Measure Theory, Integration, and Hilbert Spaces
Book 4: Functional Analysis: Introduction to Further Topics in Analysis
A series of four textbooks that aims to present, in an integrated manner,
the core areas of analysis.
Functional Analysis
- A Course in Functional Analysis by John B. Conway
- Analysis Now by Gert K. Pedersen
- Elementary Functional Analysis by Barbara MacCluer
- Introduction to Topology and Modern Analysis by George F. Simmons
- Princeton Lectures in Analysis by Elias M. Stein and Rami Shakarchi
Book 1: Fourier Analysis: An Introduction
Book 2: Complex Analysis
Book 3: Real Analysis: Measure Theory, Integration, and Hilbert Spaces
Book 4: Functional Analysis: Introduction to Further Topics in Analysis
A series of four textbooks that aims to present, in an integrated manner,
the core areas of analysis.
Ordinary Differential Equations (ODEs)
Undergraduate
- Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima
- Differential Equations with Applications and Historical Notes by George F. Simmons
- Differential Equations, Dynamical Systems, and an Introduction to Chaos by
Morris W. Hirsch, Stephen Smale, and Robert L. Devaney
Partial Differential Equations (PDEs)
Undergraduate
- Applied Partial Differential Equations with Fourier Series and Boundary
Value Problems by Richard Haberman
Graduate
- Partial Differential Equations by Lawrence C. Evans
- Partial Differential Equations: An Introduction by Walter A. Strauss
Linear Algebra
Undergraduate
- Linear Algebra by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence
- Linear Algebra Done Right by Sheldon Axler
- Matrix Analysis and Applied Linear Algebra by Carl D. Meyer
Graduate
- A Second Course in Linear Algebra by Stephan Ramon Garcia and Roger A. Horn
- Matrix Analysis by by Roger A Horn and Charles R. Johnson
- Topics in Matrix Analysis by Roger A. Horn and Charles R. Johnson
- Advanced Linear Algebra by Steven Roman
Point-Set Topology
Advanced Undergraduate / Beginning Graduate
- Topology by James Munkres
- Topology by James Dugundji
- Topology by John G. Hocking and Gail S. Young (DOVER)
- General Topology by John L. Kelley (The original intended title was "What
every young analyst should know".) (DOVER)
Algebraic Topology
Graduate
- Algebraic Topology by Allen Hatcher
- Algebraic Topology: An Introduction by William S. Massey
- Homology Theory: An Introduction to Algebraic Topology by James W. Vick
- Algebraic Topology: A First Course by William Fulton
- An Introduction to Algebraic Topology by Joseph J. Rotman
- Differential Forms in Algebraic Topology by Raoul Bott and Loring W. Tu (Combination of Algebraic and Differential Topology)
Differential Topology
Undergraduate
- Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus by Michael Spivak
- Differential Forms: A Complement to Vector Calculus by Steven H. Weintraub
- Differential Forms: Theory and Practice by Steven H. Weintraub
Graduate
- Differential Topology by Victor Guillemin and Alan Pollack
- Topology from the Differentiable Viewpoint by John Willard Milnor
- An Introduction to Differentiable Manifolds and Riemannian Geometry by William M. Boothby
- Introduction to Topological Manifolds by John Lee
- Foundations of Differentiable Manifolds and Lie Groups by Frank W. Warner
- Manifolds and Differential Geometry by Jeffrey M. Lee
- Differential Forms in Algebraic Topology by Raoul Bott and Loring W. Tu (Combination of Algebraic and Differential Topology)
Riemannian Geometry
Undergraduate
- Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo (DOVER)
Graduate
- Riemannian Geometry by Manfredo P. do Carmo
- An Introduction to Differentiable Manifolds and Riemannian Geometry by William M. Boothby
- A Comprehensive Introduction to Differential Geometry, Vol. 1, 2, 3, 4, 5 by Michael Spivak
Number Theory
Elementary Number Theory
- A Friendly Introduction to Number Theory by Joseph Silverman
- An Introduction to the Theory of Numbers by G. H. Hardy, Edward M. Wright and Andrew Wiles
- Elementary Number Theory by Gareth A. Jones and Josephine M. Jones
- Not Always Buried Deep: A Second Course in Elementary Number Theory by Paul Pollack
Algebraic Number Theory
- A Conversational Introduction to Algebraic Number Theory: Arithmetic Beyond Z by Paul Pollack
- Algebraic Number Theory by A. Frohlich and M.J. Taylor
- Algebraic Number Fields by Gerald J. Janusz
Analytic Number Theory
- Introduction to Analytic Number Theory by Tom M. Apostol
- Introduction to Analytic and Probabilistic Number Theory by Gerald Tenenbaum
- A Course in Analytic Number Theory by Marius Overholt
- Analytic Number Theory: Exploring the Anatomy of Integers by Jean-marie De Koninck and Florian Luca
- Analytic Number Theory: An Introductory Course by Paul Trevier Bateman and Harold G. Diamond
Elliptic Curves
- Elliptic Curves, Modular Forms, and Their L-functions = by Alvaro Lozano-Robledo
- The Arithmetic of Elliptic Curves by Joseph H. Silverman
- Elliptic Curves by J.S. Milne
Algebraic Geometry
Undergraduate
- Elementary Algebraic Geometry by Klaus Hulek
- Algebraic Geometry: A Problem Solving Approach by Thomas Garrity, Richard Belshoff, Lynette Boos, Ryan Brown, and Carl Lienert
- Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and
Commutative Algebra by David A. Cox, John Little, and Donal O'Shea
Graduate
- Commutative Algebra: with a View Toward Algebraic Geometry by David Eisenbud
- Algebraic Geometry by Robin Hartshorne
- Principles of Algebraic Geometry by Phillip Griffiths and Joseph Harris
- Rational Points on Varieties by Bjorn Poonen
Logic, Set Theory, and Computability
Undergraduate
- A Mathematical Introduction to Logic by Herbert Enderton
- The Joy of Sets: Fundamentals of Contemporary Set Theory by Keith Devlin
- Axiomatic Set Theory by Patrick Suppes (DOVER)
- A Book of Set Theory by Charles C. Pinter (DOVER)
- Computability Theory by Rebecca Weber
Graduate
- Introduction to Mathematical Logic by Elliott Mendelson
- Logic for Mathematicians by A.G. Hamilton
- Set Theory by Thomas Jech (Some readers suggest the 1978 version is more suitable for beginners than the Millennium Edition.)
- Set Theory by Kenneth Kunen
- Godel's Proof by Ernest Nagel and James Newman
- Set Theory and the Continuum Hypothesis by Paul J. Cohen
- Computability: An Introduction to Recursive Function Theory by Nigel Cutland
- Computability: A Mathematical Sketchbook by Douglas S. Bridges
Category Theory
- Categories for the Working Mathematician by Saunders Mac Lane
- Category Theory in Context by Emily Riehl (DOVER)
- Basic Category Theory by Tom Leinster
Copyright 2020.
Maintained by
Mark Tomforde.