Algebra

Abstract Algebra

• A Course in Universal Algebra - Free online edition of Stanley Burris and H. P. Sankappanavar's graduate text on universal algebra.
• Elements of Abstract and Linear Algebra - E.H. Connell (University of Miami).
• Algebraic Methods - F. Oggier.
• Introduction to Modern Algebra - David Joyce (Clark University).
• Abstract Algebra: Theory and Applications - Thomas W. Judson.
• Abstract Algebra: Harvard Extension - Benedict Gross.

Commutative Algebra

• The CRing Project - Open-source work-in-progress textbook introducing commutative algebra for readers with elementary abstract algebra, with an emphasis on foundations for algebraic geometry.
• Commutative Algebra Notes - Free graduate course notes by Mel Hochster, from an introductory course through advanced topics such as Cohen-Macaulay rings, multiplicities, and étale maps.

Formal Mathematics and Theorem Proving

• Elementary Algebra 2e - Free one-semester textbook from OpenStax with web and PDF editions, worked examples, practice exercises, chapter reviews, and answer keys.

Group Theory

• Group Theory - J.S. Milne.
• Notes on Finite Group Theory - Peter J. Cameron.
• Group Theory - Predrag Cvitanović.

Linear Algebra

• Fundamentals of Linear Algebra - James B. Carrell.
• Linear Algebra - David Cherney, Tom Denton, Andrew Waldron.
• Introduction to Vectors and Tensors, Vol. 1: Linear and Multilinear Algebra - Ray M. Bowen, C. C. Wang.
• Introduction to Vectors and Tensors, Vol. 2: Vector and Tensor Analysis - Ray M. Bowen, C. C. Wang.
• Introduction to Applied Linear Algebra - Stephen Boyd (Stanford University), Lieven Vandenberghe (UCLA).
• Interactive Linear Algebra - Dan Margalit and Joseph Rabinoff.
• MIT OpenCourseWare Lectures on Linear Algebra (18.06) as Jupyter Notebooks - Juan Klopper.
• Linear Algebra Done Right - Sheldon Axler.
• Linear Algebra - Jim Hefferon.
• A First Course in Linear Algebra - Robert A. Beezer.
• Immersive Linear Algebra - J. Ström, K. Åström, and T. Akenine-Möller.

Representation Theory

• Lie Algebras - Shlomo Sternberg.
• Introduction to Representation Theory - Free MIT course by Pavel Etingof with lecture notes and problem sets on representations of groups and algebras, categories, and quivers.

Ring and Field Theory

• Surreal Numbers - How two ex-students turned on to pure mathematics and found total happiness - D. E. Knuth.
• Surreal Numbers and Games - Introductory notes on Conway's surreal numbers and their connection with combinatorial game theory.
• Foundations of Module and Ring Theory - Robert Wisbauer (University of Düsseldorf).
• Fields and Galois Theory - J.S. Milne.
• Galois Theory - Ian Stewart.
• Galois Theory - Tom Leinster (University of Edinburgh).
• An Introduction to Galois Theory - Andrew Baker (University of Glasgow).
• Galois Theory - Miles Reid.
• Conway names, the simplicity hierarchy and the surreal number tree - Philip Ehrlich.

Analysis

Complex Analysis

• Introduction to Complex Analysis - Michael Taylor.
• Complex Variables - R. B. Ash, W.P. Novinger.
• Complex Analysis - Christer Bennewitz.
• A Concise Course in Complex Analysis and Riemann Surfaces - Wilhelm Schlag.
• Complex Analysis - G. Cain (Georgia Tech).
• Complex Analysis - Juan Carlos Ponce Campuzano.
• An Introduction to Complex Analysis and Geometry - Paid textbook by John P. D'Angelo.
• A First Course in Complex Analysis - Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka.
• A Guide to Complex Variables - Paid textbook by Steven G. Krantz.
• Complex Analysis - Charles Walkden.
• Complex Analysis - Donald E. Marshall.
• Complex Analysis - Christian Berg.

Functional Analysis

• Functional Analysis - Alexander C. R. Belton.
• Theory of Functions of a Real Variable - Shlomo Sternberg.
• Functional Analysis - Lawerence Baggett.
• An Introduction to Functional Analysis - Laurent W. Marcoux (University of Waterloo).
• Functional Analysis - Christian Remling.
• Functional Analysis: Lecture Notes - Jeff Schenker (Michigan State University).

Harmonic Analysis

• Lecture Notes: Fourier Transform and its Applications - Brad Osgood.
• Harmonic Analysis Lecture Notes - Richard S. Laugesen (University of Illinois at Urbana-Champaign).

Measure Theory

• Measure Theory - John K. Hunter (University of California at Davis).
• Measure and Integration - Dietmar A. Salamon (ETH Zürich).
• A Crash Course on the Lebesgue Integral and Measure Theory - Steve Cheng.
• Lecture Notes: Measure Theory - Bruce K. Driver.
• Lecture Notes in Measure Theory - Christer Borell.

Operator Theory

• Lecture Notes on Operator Algebras - Free graduate notes by John M. Erdman on Hilbert-space operators, Banach algebras, spectral theory, C*-algebras, compact operators, and K-theory.

Real Analysis

• An Introduction to Real Analysis - John K. Hunter (University of California at Davis).
• Introduction to Real Analysis - William F. Trench (Trinity University, Texas).
• Basic Analysis: Introduction to Real Analysis - Jiří Lebl.
• Lecture Notes in Real Analysis - Eric T. Sawyer (McMaster University).
• Modern Real Analysis - William P. Ziemer (Indiana University).
• Analysis of Functions of a Single Variable - Lawerence Baggett.
• The Calculus of Functions of Several Variables - Dan Sloughter.
• A ProblemText in Advanced Calculus - John M. Erdman.
• Calculus and Linear Algebra. Vol. 2 - Wilfred Kaplan, Donald J. Lewis.
• Active Calculus - Matt Boelkins.
• Supplements to the Exercises in Chapters 1-7 of Walter Rudin's "Principles of Mathematical Analysis" - George M. Bergman.
• Calculus Made Easy - Silvanus P. Thompson (1910).
• Precalculus - Carl Stitz, Jeff Zeager.
• Calculus Open Textbook - Gilbert Strang (MIT OpenCourseWare).
• Elementary Calculus: An Approach Using Infinitesimals - Professor H. Jerome Keisler.
• Elementary Real Analysis - Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner.
• Advanced Real Analysis - Curtis T. McMullen.
• Real Analysis for Graduate Students - Richard F. Bass.
• Advanced Calculus - Lynn Loomis, Schlomo Sternberg.
• Calculus and Linear Algebra. Vol. 1 - Wilfred Kaplan, Donald J. Lewis.

Combinatorics and Discrete Mathematics

Analytic Number Theory

• Basic Combinatorics - Carl G. Wagner (University of Tennessee).
• Notes on Combinatorics - Peter J. Cameron.
• Analytic Combinatorics - Philippe Flajolet, Robert Sedgewick.
• Applied Combinatorics - Mitchel T. Keller, William T. Trotter.
• Discrete Mathematics: An Open Introduction - Free fourth-edition undergraduate textbook by Oscar Levin, with inquiry activities, more than 750 exercises, solutions and hints, and coverage of proofs, graph theory, counting, sequences, and discrete structures.
• generatingfunctionology - Herbert Wilf.
• A=B - Marko Petkovšek, Herbert Wilf, and Doron Zeilberger on algorithms for proving hypergeometric identities.

Graph Theory

• Graph Theory - Reinhard Diestel.
• Graph Theory : Interactive Algorithm Visualizer | Graph Theory Learning Platform - Hadjoudj Mohammed Islam.
• Graph Theory: Lecture Notes - Christopher Griffin.

Community and Further Reading

Blogs

• BetterExplained - Maintained by Kalid Azad.
• 3Blue1Brown - Animated explanations of mathematics by Grant Sanderson.
• ILoveMaths - Mathematics lessons and activities for Indian classes 6 to 12.
• Math Is Fun - Lightweight lessons and reference pages for school mathematics.

Journals and Magazines

• Quanta Magazine - Features latest research breakthroughs in an accessible style for non-experts.
• European Mathematical Society Magazine - The Magazine features announcements about meetings and conferences, articles outlining current trends in scientific development, reports on member societies, and many other informational items.
• Mathematics Today by Institute of Mathematics and its Applications - News, opinions, and articles related to mathematics, so the reader stays updated.
• Crux Mathematicorum by Canadian Mathematical Society - Challenging problems for secondary and undergraduate students, including an Olympiad Corner.
• Notices of the American Mathematical Society - Publicizes activities of the Society and features surveys, reports, news, announcements, and opinions on industry trends, academia, and research.
• Chalkdust Magazine - Independent magazine for mathematically curious readers, with free expository articles, interviews, puzzles, and downloadable issues.
• Bulletin of the American Mathematical Society - Expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic.

Meetings and Conferences

• MathsJam - Monthly local recreational mathematics and puzzle meetups, plus an annual gathering in the United Kingdom.
• Talking Maths in Public - Biennial conference for mathematics communicators in the United Kingdom.
• Bridges - Annual conference on mathematical connections in art, music, architecture, and culture.

Related Curated Lists

• Theoretical Computer Science - Curated resources on computation, complexity, logic, algorithms, cryptography, and related mathematics.
• Scientific Computing - Curated software for numerical analysis, scientific computing, meshing, solvers, and visualization.
• Optimization - Courses, books, notes, and software across mathematical optimization and operations research.
• Interactive Mathematics - Tools and libraries for building interactive mathematical explanations and explorations.
• AI for Mathematics - Research index covering AI-assisted mathematical reasoning, discovery, formal proof, and related datasets. Individual entries require independent review.

Differential Equations and Dynamical Systems

Ergodic Theory and Chaos

• Chaos: Classical and Quantum - Maintained technical webbook on dynamical systems, periodic orbits, deterministic chaos, statistical mechanics, and quantum chaos.

Ordinary Differential Equations

• Ordinary Differential Equation - Alexander Grigorian (University of Bielefeld).
• Ordinary Differential Equations: Lecture Notes - Eugen J. Ionascu.
• Ordinary Differential Equations and Dynamical Systems - Gerald Teschl.
• Elementary Differential Equations - William F. Trench.
• Elementary Differential Equations With Boundary Value Problems - William F. Trench.
• Notes on Diffy Qs: Differential Equations for Engineers - Jiří Lebl.
• Differential Equations - Gabriel Nagy.

Partial Differential Equations

• Notes on Partial Differential Equations - John K. Hunter (University of California at Davis).
• Linear Methods of Applied Mathematics - Evans M. Harrell II, James V. Herod (Georgia Tech).
• Partial Differential Equations - Lawrence C. Evans's concise survey of modern PDE theory from analytical, qualitative, and computational viewpoints.

Foundations and Logic

Category Theory

• Introduction to Category Theory and Categorical Logic - Thomas Streicher.
• Category Theory - B. Pareigis.
• Toposes, Triples and Theories - Michael Barr, Charles Wells.
• Abelian Categories - Peter Freyd.
• Categories and Groupoids - P. J. Higgins.
• Basic Concepts of Enriched Category Theory - G. M. Kelly.
• Abstract and Concrete Categories: The Joy of Cats - Jiri Adamek, Horst Herrlich, George E. Strecker.
• Category Theory in Context - Emily Riehl (Johns Hopkins University).
• Category Theory - Paid textbook by Steve Awodey.
• Category Theory for Computing Science - Michael Barr, Charles Wells.
• Seven Sketches in Compositionality: An Invitation to Applied Category Theory - Brendan Fong and David I. Spivak (MIT).

Formal Mathematics and Theorem Proving

• Lean - Open-source interactive theorem prover and programming language based on dependent type theory, used for formal mathematics and software verification.
• mathlib - Community-maintained mathematical library for Lean 4, with formalized theories, proof tactics, programming infrastructure, and generated documentation.
• Mathematics in Lean - Free textbook with examples and exercises for mathematicians learning Lean 4 and mathlib.
• Natural Number Game - Interactive Lean 4 game that introduces theorem proving through guided exercises about natural numbers.
• Isabelle - General-purpose interactive theorem prover with higher-order logic and set-theory environments for formalizing mathematics and computer science.
• Rocq Prover - Open-source interactive theorem prover and dependently typed programming language for mechanized mathematics and verified software.
• Mathematical Components - Rocq libraries and tools for large-scale formalized mathematics, including substantial developments in algebra and analysis.
• Metamath - Minimal formal language and proof verifier with explicit, inspectable proofs built from simple foundations.
• Courses Using Lean - Maintained, filterable index of university courses that teach Lean or use it for mathematics, logic, programming, and formal verification.
• Lean-related Conferences and Events - Maintained calendar and archive of conferences, workshops, and tutorials about Lean, mathlib, and formalized mathematics.
• Theorem Proving in Lean 4 - Free official textbook on dependent type theory, propositions, tactics, inductive types, type classes, and constructing verified proofs in Lean 4.
• Archive of Formal Proofs - Refereed collection of Isabelle proof libraries, examples, and scientific developments maintained against current Isabelle releases.

Logic

• An Introduction to Formal Logic - P.D. Magnus (University at Albany).
• Computability - An introduction to recursive function theory - Nigel Cutland (University of Hull).
• Formal Logic - Miguel Palomino.
• Predictive Arithmetic - Edward Nelson.
• Proofs and Concepts: the fundamentals of abstract mathematics - Joy Morris, Dave Morris.
• Open Logic Textbook - collaborative effort, main contributors listed here.
• forall x: Calgary (An Introduction to Formal Logic) - P.D. Magnus and Tim Button, remixed by Aaron Thomas-Bolduc and Richard Zach (Open Logic Project).
• Introduction to Logic - Michael Genesereth, Eric J. Kao (Stanford University).
• Language, Proof, and Logic, Second Edition - Paid textbook by Dave Barker-Plummer, Jon Barwise, and John Etchemendy.
• Mathematical Logic - Helmut Schwichtenberg.
• Mathematical Logic - Stephen G. Simpson (Pennsylvania State University).
• Mathematical Reasoning: Writing and Proof - Ted Sundstrom.
• Logic and Proof - Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn.

Set Theory

• Sets, Relations, Functions - Ivo Düntsch, Günther Gediga.
• Set Theory and Foundations of Mathematics - Sylvain Poirier.
• Set Theory on the Stanford Encyclopedia of Philosophy - Peer-reviewed overview of set theory with historical context and references.
• An Introduction to Set Theory - William A. R. Weiss.

Type Theory

• Proofs and Types - Jean-Yves Girard.
• Intuitionistic Type Theory - Per Martin-Lof.
• Type Theory and Functional Programming - Simon Thompson.
• Programming in Martin-Lof’s Type Theory - Bengt Nordstrom, Kent Petersson, Jan M. Smith.

General Mathematics

Broad Course Notes

• Paul's Online Math Notes - Free notes, examples, and practice problems for algebra, calculus, and differential equations, by Paul Dawkins at Lamar University.
• UBC CLP Calculus Textbooks - Free textbooks, problem books, and source files for a four-course university sequence in differential, integral, multivariable, and vector calculus.

Geometry and Topology

Algebraic Geometry

• A Brief Introduction to Algebraic Geometry - R.C. Churchill.
• Foundations of Algebraic Geometry - Ravi Vakil.
• Algebraic Geometry - Jean Gallier, Stephen S. Shatz (University of Pennsylvania).
• Algebraic Geometry - J.S. Milne.
• The Stacks Project - Maintained by Aise Johan de Jong (Columbia).
• Introduction to Algebraic Geometry - Igor V. Dolgachev.
• Algebraic Geometry - Andreas Gathmann (RPTU).

Algebraic Topology

• Algebraic Topology - Allen Hatcher.
• A Concise Course in Algebraic Topology - J. P. May.
• Introduction to Algebraic Topology - Martin Cadek.
• Algebra and Topology - Pierre Schapira (Sorbonne University).
• Lecture Notes in Algebraic Topology - Paid textbook by James F. Davis and Paul Kirk.

Differential Geometry

• Introduction to Differential Geometry - Joel W. Robbin, Dietmar A. Salamon.
• Topics in Differential Geometry - Peter W. Michor.
• Lectures on Differential Geometry - Wulf Rossmann.
• Semi-Riemann Geometry and General Relativity - Shlomo Sternberg.
• Discrete Differential Geometry - Keenan Crane.
• Notes on Differential Geometry and Lie Groups - Jean Gallier (University of Pennsylvania).
• An Introduction to Riemannian Geometry - Sigmundur Gudmundsson (Lund University).
• Differential Geometry - Rigorous undergraduate MIT OpenCourseWare course with lecture notes and problem sets centered on curves, surfaces, and curvature.
• The Geometry and Topology of Three-Manifolds - W. Thurston.

General Topology

• Introduction to Topology - Free introductory notes associated with Renzo Cavalieri's Colorado State University topology course and compiled by students. Covers point-set topology, compactness, connectedness, surfaces, Euler characteristic, and the fundamental group.
• Introduction to Topology - Alex Küronya.
• General Topology - Jesper M. Møller.
• Elementary Applied Topology - Robert Ghrist (UPenn).
• General Topology - Stephen Willard.
• General Topology - Pierre Schapira (Sorbonne University).
• Elementary Topology Problem Textbook - Oleg Viro.

Graph Theory

• Fundamentals of Geometry - Free geometry text by Oleg A. Belyaev.
• Mathematical Illustrations - Bill Casselman's guide to creating clear mathematical diagrams and illustrations.
• Byrne's Euclid - Interactive web edition of Oliver Byrne's color-coded presentation of Euclid's first six books.
• Euclid's Elements (David Joyce edition) - Complete web edition with diagrams, commentary, and references by David Joyce.
• Euclid's Elements Redux - Open textbook by Daniel Callahan that presents Euclid with modern commentary and exercises.

History and Mathematics Education

Mathematical Software and Tools

• MacTutor History of Mathematics Archive - University of St Andrews archive of mathematician biographies, histories of mathematical topics, timelines, institutions, and related reference material.