Foundations and Logic / Logic

• 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).

Geometry and Topology / General Topology

• General Topology - Stephen Willard.

Start Here / Video Courses and Series

• Numberphile - Popular mathematics videos featuring mathematicians discussing problems, ideas, and curiosities.

Start Here / Reference Works

• The Math Behind Artificial Intelligence - Tiago Capelo Monteiro (freeCodeCamp).

Combinatorics and Discrete Mathematics / Graph Theory

• Graph Theory : Interactive Algorithm Visualizer | Graph Theory Learning Platform - Hadjoudj Mohammed Islam.

Mathematical Practice / Mathematical Software and Tools

• MathFlow (⭐56) - C# math expression library with symbolic computation (differentiation, simplification, equation solving).

• RunMat (⭐247) - Runtime for MATLAB-syntax array math with automatic CPU/GPU execution.

Algebra / Abstract Algebra

• Algebraic Methods - F. Oggier.

Number Theory / Algebraic Number Theory

• Introduction to Algebraic Number Theory - F. Oggier.

Algebra / Linear Algebra

• Linear Algebra Done Right - Sheldon Axler.

Start Here / Proof and Problem Solving

• Book of Proof - Richard Hammack (Virginia Commonwealth University).

Geometry and Topology / Differential Geometry

• Notes on Differential Geometry and Lie Groups - Jean Gallier (University of Pennsylvania).

Start Here / Learning Platforms

• Math Academy - Paid adaptive platform with diagnostic assessment and mastery-based courses from fourth-grade mathematics through university topics.

Community and Further Reading / 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.

Algebra / Ring and Field Theory

• Galois Theory - Ian Stewart.

• Galois Theory - Tom Leinster (University of Edinburgh).

Numerical and Computational Mathematics / Numerical Analysis

• Advanced Linear Algebra: Foundations to Frontiers - Online text by Robert van de Geijn and Margaret Myers on algorithms for advanced numerical linear algebra.

Start Here / Reference Works

• The Princeton Companion to Mathematics - Paid reference edited by Timothy Gowers, June Barrow-Green, and Imre Leader.

• Encyclopedia of Distances (4th Edition) - Paid reference by Michel Marie Deza and Elena Deza.

Analysis / Functional Analysis

• An Introduction to Functional Analysis - Laurent W. Marcoux (University of Waterloo).

Numerical and Computational Mathematics / Numerical Analysis

• Advanced Numerical Methods and Their Applications to Industrial Problems: Adaptive Finite Element Methods - Specialized notes by Alfred Schmidt and Arsen Narimanyan on adaptive finite element methods.

Number Theory / Representation Theory

• Yet Another Introductory Number Theory Textbook - Jonathan A. Poritz.

Probability and Statistics / Algebraic Statistics

• Lectures on Algebraic Statistics - Mathias Drton, Bernd Sturmfels, Seth Sullivant.

• An Introduction to Algebraic Statistics - Cristiano Bocci, Luca Chiantini and Anthony V. Geramita.

• Algebraic Statistics - Karl-Heinz Zimmermann.

Start Here / Reference Works

• The On-Line Encyclopedia of Integer Sequences - Great compendium of many different integer sequences. Founded 1964 by N. J. A. Sloane.

Start Here / Video Courses and Series

• StatQuest - Introductory video explanations of statistics, probability, and machine learning.

Start Here / Video Courses and Series

• Brandon Foltz - Video courses in statistics, probability, and quantitative methods with worked examples.

• 3Blue1Brown - Visual explanations of core mathematics, including calculus, linear algebra, and differential equations.

• NPTEL - University lecture courses from Indian institutes covering mathematics and related technical subjects.

• PatrickJMT - Worked-example videos for algebra, calculus, and other introductory college topics.

• Professor Leonard - Full lecture series in precalculus, calculus, and differential equations for college students.
  • Precalculus - College Algebra/Trigonometry - Full course playlist.
  • Calculus 1 - Full course playlist.
  • Calculus 2 - Full course playlist.
  • Calculus 3 - Full course playlist.
  • Differential Equations - Full course playlist.
  • To The Point Math - Short reviews of common algebra and calculus topics.

• MIT OpenCourseWare - MIT course recordings and video lectures, including full mathematics courses.

• Mathologer - Visual videos on recreational mathematics and selected advanced topics.

Analysis / Measure Theory

• An Introduction to Measure Theory - Terence Tao (UCLA).

Numerical and Computational Mathematics / Numerical Analysis

• Numerical Analysis for Engineers - University of Waterloo course with lectures, assignments, projects, and MATLAB material on numerical methods for engineers.

Start Here / Learning Platforms

• Oxford Mathematics - Lectures and public talks from the University of Oxford, including accessible explanations of advanced topics.

Community and Further Reading / Related Curated Lists

• Theoretical Computer Science (⭐1.2k) - Curated resources on computation, complexity, logic, algorithms, cryptography, and related mathematics.

Community and Further Reading / 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.

Start Here / Learning Platforms

• Almost Fun - Free interactive middle- and high-school mathematics lessons, practice problems, and teacher resources from a nonprofit organization.

Geometry and Topology / Algebraic Geometry

• A Brief Introduction to Algebraic Geometry - R.C. Churchill.

Start Here / Learning Platforms

• Ximera - Free interactive mathematics textbooks from Ohio State University.

Analysis / Complex Analysis

• A Concise Course in Complex Analysis and Riemann Surfaces - Wilhelm Schlag.

Mathematical Practice / Mathematical Software and Tools

• Maxima - Free computer algebra system for symbolic and numerical calculations, plotting, and scripting.

• Free Math - Free, open-source software for recording, organizing, and reviewing digital mathematics homework without solving the problems automatically.

Analysis / Complex Analysis

• Introduction to Complex Analysis - Michael Taylor.

Start Here / Broad Course Notes

• Evan Chen - MIT. 2012 ~ 2018. Covers Combinatorics, Number Theory, Honors Algebra, Set Theory, Real Analysis, Graph Theory, and more.

• Dexter Chua - Harvard. 2013 ~ 2018. Covers Analysis, Probability, Linear Algebra, Complex Analysis, Numerical Analysis, Statistics, Optimization, Algebraic Topology, Quantum Field Theory, and more.

Analysis / Real Analysis

• Precalculus - Carl Stitz, Jeff Zeager.

Analysis / Harmonic Analysis

• Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications - Julius O. Smith III (Stanford University).

Mathematical Practice / Mathematical Software and Tools

• GeoGebra - Interactive mathematics suite for geometry, graphing, algebra, spreadsheets, statistics, and calculus.

• Singular - Computer algebra system for polynomial computations in commutative algebra, algebraic geometry, and singularity theory.

• Maple - Paid computer algebra and numerical mathematics system for symbolic computation, visualization, and programming.

• Wolfram Mathematica - Paid technical-computing system for symbolic and numerical mathematics, visualization, programming, and data analysis.

Start Here / Learning Platforms

• Mathigon - Interactive courses, activities, and visual explanations for school and early university mathematics.

Foundations and Logic / Logic

• QED - an interactive textbook - Terence Tao.

Algebra / Linear Algebra

• Interactive Linear Algebra - Dan Margalit and Joseph Rabinoff.

Geometry and Topology / Graph Theory

• Byrne's Euclid - Interactive web edition of Oliver Byrne's color-coded presentation of Euclid's first six books.

Analysis / Complex Analysis

• Complex Analysis - Juan Carlos Ponce Campuzano.

Probability and Statistics / Probability

• Foundations of Constructive Probability Theory - Yuen-Kwok Chan.

Probability and Statistics / Statistics

• Statistical Thinking for the 21st Century - Russell A. Poldrack.

• Odds and Ends: Introducing Probability & Decision with a Visual Emphasis - Jonathan Weisberg.

• Seeing Theory - Daniel Kunin, Jingru Guo, Tyler Dae Devlin, and Daniel Xiang.

• Statistics Done Wrong - Alex Reinhart.

• All of Statistics: A Concise Course in Statistical Inference - Larry Wasserman.

Optimization, Operations Research, and Control / Numerical Analysis

• Convex Optimization - A free textbook with slides, exercises, code, and course material by Stephen Boyd and Lieven Vandenberghe.

Interdisciplinary Mathematics / Mathematics for Computer Science

• The Chaos Textbook: Mathematics in the age of the computer - Glenn Elert.

Interdisciplinary Mathematics / Signal Processing

• Think DSP - Allen B. Downey.

Foundations and Logic / Type Theory

• Intuitionistic Type Theory - Per Martin-Lof.

• Type Theory and Functional Programming - Simon Thompson.

Algebra / Abstract Algebra

• Abstract Algebra: Harvard Extension Videos - Benedict Gross.

Algebra / Linear Algebra

• Linear Algebra Done Wrong - Sergei Treil.

Analysis / Real Analysis

• Supplements to the Exercises in Chapters 1-7 of Walter Rudin's "Principles of Mathematical Analysis" - George M. Bergman.

Start Here / Learning Platforms

• Brilliant - Interactive, problem-based lessons in foundational mathematics, probability, and related quantitative subjects.

• WooTube - Video lessons by Eddie Woo covering secondary-school mathematics with an emphasis on conceptual understanding.

Geometry and Topology / General Topology

• Elementary Applied Topology - Robert Ghrist (UPenn).

Geometry and Topology / Algebraic Geometry

• The Stacks Project - Maintained by Aise Johan de Jong (Columbia).

Mathematical Practice / Mathematical Software and Tools

• Symbolab - Online solver with step-by-step solutions, graphing, calculators, and practice tools. Some solution steps and study features require a paid plan.

• Desmos - Browser-based graphing calculator with geometry, tables, statistics, and classroom activities.

Algebra / Linear Algebra

• Introduction to Applied Linear Algebra - Stephen Boyd (Stanford University), Lieven Vandenberghe (UCLA).

Foundations and Logic / Logic

• Predictive Arithmetic - Edward Nelson.

Algebra / Linear Algebra

• Linear Algebra - David Cherney, Tom Denton, Andrew Waldron.

Analysis / Real Analysis

• Calculus and Linear Algebra. Vol. 2 - Wilfred Kaplan, Donald J. Lewis.

Combinatorics and Discrete Mathematics / Analytic Number Theory

• A=B - Marko Petkovšek, Herbert Wilf, and Doron Zeilberger on algorithms for proving hypergeometric identities.

Probability and Statistics / Stochastic Processes

• Introduction to Stochastic Processes - Lecture Notes - Gordan Žitković (University of Texas).

• Applied Stochastic Processes in science and engineering - Matt Scott (University of Waterloo).

Interdisciplinary Mathematics / Signal Processing

• An Introduction to Statistical Signal Processing - Robert M. Gray, Lee D. Davisson.

Community and Further Reading / Blogs

• BetterExplained - Maintained by Kalid Azad.

Foundations and Logic / Logic

• An Introduction to Formal Logic - P.D. Magnus (University at Albany).

Probability and Statistics / Probability

• Probability and Statistics Cookbook (⭐2.3k) - Matthias Vallentin (UC Berkeley).

Start Here / Reference Works

• ProofWiki - Collaborative repository of mathematical definitions, lemmas, and proofs.

Algebra / Linear Algebra

• MIT OpenCourseWare Lectures on Linear Algebra (18.06) as Jupyter Notebooks (⭐585) - Juan Klopper.

Start Here / Learning Platforms

• Khan Academy - Free, self-paced lessons and exercises from arithmetic through calculus and beyond.

• Coursera - University and industry mathematics courses, with access and payment varying by course.

• edX - University and professional mathematics courses, with audit and paid options varying by course.

Start Here / Reference Works

• Wikibooks - Community-authored collection of open mathematics textbooks and instructional books.

Foundations and Logic / Type Theory

• Homotopy Type Theory - Open textbook on homotopy type theory, univalent foundations, and higher-dimensional structures.

Geometry and Topology / Differential Geometry

• Introduction to Differential Geometry - Joel W. Robbin, Dietmar A. Salamon.

Analysis / Real Analysis

• An Introduction to Real Analysis - John K. Hunter (University of California at Davis).

Analysis / Harmonic Analysis

• Lecture Notes: Fourier Transform and its Applications - Brad Osgood.

Analysis / Measure Theory

• Measure Theory - John K. Hunter (University of California at Davis).

• Measure and Integration - Dietmar A. Salamon (ETH Zürich).

Differential Equations and Dynamical Systems / Ordinary Differential Equations

• Ordinary Differential Equation - Alexander Grigorian (University of Bielefeld).

Differential Equations and Dynamical Systems / Partial Differential Equations

• Notes on Partial Differential Equations - John K. Hunter (University of California at Davis).

Probability and Statistics / Statistical Learning

• Statistical Learning Theory - Percy Liang.