Awesome Math

A curated list of awesome mathematics resources.

Contents

• Contents
• General Resources
  • Learning Platforms
  • Learn to Learn
  • Youtube Series
  • Tools
  • Questions and Answers
  • Encyclopedia
  • Books
  • Blogs
  • Misc
• Lecture Notes
  • Foundations of Mathematics
    • Transition To Pure Rigour Math
    • Set Theory
    • Logic
    • Category Theory
    • Type Theory
    • Homotopy Type Theory
    • Surreal Numbers
  • Number Theory
    • Algebraic Number Theory
    • Analytic Number Theory
  • Algebra
    • Abstract Algebra
    • Group Theory
    • Linear Algebra
    • Ring Theory
    • Galois Theory
    • Lie Algebras
  • Combinatorics
    • Graph Theory
  • Geometry and Topology
    • Differential Geometry
    • Algebraic Geometry
    • Topology
    • Algebraic Topology
  • Analysis
    • Real Analysis
    • Harmonic Analysis
    • Complex Analysis
    • Functional Analysis
    • Measure Theory
    • Ordinary Differential Equations
    • Partial Differential Equations
  • Probability and Statistics
    • Probability Theory
    • Statistics
    • Statistical Learning
    • Stochastic processes
  • Numerical Analysis
  • Signal processing
  • Mathematics for Computer Science
  • Mathematical Biology
  • Mathematical Physics
• Students Lecture Notes
• License

General Resources

Learning Platforms

• Khan Academy
• Coursera
• MIT OpenCourseWare
• edX
• Brilliant
• WooTube
• Mathigon
• Calculus.org
• Ximera : free interactive mathematics textbooks (Ohio State University)

Learn to Learn

• Understanding Mathematics

Youtube Series

• Brandon Foltz
• 3blue1brown
• NPTEL
• Prof. Leonard
• Crash Course
• Harvard
• MIT

Tools

• Symbolab
• Desmos
• Math Words
• Wolfram Alpha
• Maxima
• Sympy
• Sagemath
• Unit Converter
• GeoGebra
• Macaulay2
• Singular
• GNU Octave
• Magma
• Maple
• Matlab
• Wolfram Mathematica
• Free Math

Questions and Answers

• Mathematics Stack Exchange
• MathOverflow - for professional mathematicians

Encyclopedia

• Encyclopedia of Mathematics
• Planetmath
• ProofWiki
• Wolfram Mathworld

Books

• Calculus: Basic Concepts for High Schools - L.V. Tarasov
• Basics of Algebra, Topology, and Differential Calculus - Jean Gallier (University of Pennsylvania)
• Multivariable Calculus - G. Cain, J. Herod (Georgia Tech)
• Wikibooks
• Online Mathematics Textbooks
• Beginning and Intermediate Algebra
• Free Mathematics Books
• Trigonometry

Blogs

• BetterExplained - Maintained by Kalid Azad
• ILoveMaths - For grades 6 thru 12 in K-12 system
• 3blue1brown - Animated Maths
• Mathsisfun simple text light weight site for students upto highschool
• MathematicsIsAScience - Peter Saveliev (Professor of mathematics at Marshall University, Huntington WV, USA)

Misc

• Areas of mathematics on Wikipedia
• Paul's Online Math Notes - Paul Dawkins (Lamar University)
• List of electronic textbooks - Marcel B. Finan (Arkansas Tech University)
• Topology Atlas
• Recreations in Math - H. E. Licks (1917)
• Magic Squares and Cubes - W. S. Andrews (1917)
• Convex Optimization - Stephen Boyd and Lieven Vandenberghe

Lecture Notes

Foundations of Mathematics

Transition To Pure Rigour Math

• Basic Concepts of Mathematics - Elias Zakon
• Book of Proof - Richard Hammak (Virginia Commonwealth University)

Set Theory

• Sets, Relations, Functions - Ivo Düntsch, Günther Gediga
• An Introduction to Set Theory - William A. R. Weiss
• Set Theory and Foundations of Mathematics - Sylvain Poirier
• Set Theory on the Stanford Encyclopedia of Philosophy

Logic

• Introduction to Logic - Michael Genesereth, Eric Kao (Stanford University)
• An Introduction to Formal Logic - P.D. Magnus (University at Albany)
• A Problem Course in Mathematical Logic - Stefan Bilaniuk (Trent University)
• Language, Proof, and Logic - Jon Barwise, John Etchemendy
• Mathematical Logic - Helmut Schwichtenberg
• Mathematical Logic - Stephen G. Simpson (Pennsylvania State University)
• Formal Logic - Miguel Palomino
• Predictive Arithmetic - Edward Nelson
• Proofs and Concepts: the fundamentals of abstract mathematics - Joy Morris, Dave Morris
• Mathematical Reasoning: Writing and Proof - Ted Sundstrom
• Logic and Proof - Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn
• QED - an interactive textbook - Terence Tao
• Open Logic Textbook - collaborative effort, main contributors listed here

Category Theory

• Introduction to Category Theory and Categorical Logic - Thomas Streicher
• An Introduction to Category Theory - Harold Simmons
• Category Theory - Steve Awodey (Carnegie Mellon University)
• Category Theory - B. Pareigis
• Category Theory for Computing Science - Michael Barr, Charles Wells
• 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. Kelley
• Abstract and Concrete Categories: The Joy of Cats - Jiri Adamek, Horst Herrlich, George Strecker
• Seven Sketches in Compositionality: An Invitation to Applied Category Theory - Brendan Fong and David I. Spivak (MIT)
• Category Theory in Context - Emily Riehl (John Hopkins University)

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

Homotopy Type Theory

• Homotopy Type Theory

Surreal Numbers

• Surreal Numbers - How two ex-students turned on to pure mathematics and found total happiness - D. E. Knuth
• An Introduction to Surreal Numbers - Gretchen Grimm
• Surreal Numbers and Games
• Conway names, the simplicity hierarchy and the surreal number tree - Philip Ehrlich

Number Theory

• Elementary Number Theory: Primes, Congruences, and Secrets - William Stein
• Elementary Number Theory - W. Edwin Clark (University of South Florida)
• A Course on Number Theory - Peter J. Cameron
• A Computational Introduction to Number Theory and Algebra - Victor Shoup
• Number Theory: A Contemporary Introduction - Pete L. Clark
• An Introduction to the Theory of Numbers - Leo Moser
• Yet Another Introductary Number Theory Textbook - Jonathan A. Poritz
• Elementary Number Theory - David M. Burton

Algebraic Number Theory

• Introduction to Algebraic Number Theory
• Introduction to Algebraic Number Theory - F. Oggier
• Algebraic Number Theory - J.S. Milne
• Algebraic Number Theory Course Notes - Matthew Baker (Georgia Tech)
• A Course In Algebraic Number Theory - Robert Ash

Analytic Number Theory

• Introduction to Analytic Number Theory - A.J. Hildebrand (University of Illinois)
• Elements of Analytic Number Theory - P. S. Kolesnikov, E. P. Vdovin (Novosibirsk)
• Analytic Number Theory - Otto Forster (LMU Munich)
• Analytic Number Theory - Lecture Notes based on Davenport’s book - Andreas Strömbergsson

Algebra

• A Course in Universal Algebra - S. Burris, H.P. Sankappanavar
• A Course in Commutative Algebra - Robert Ash
• First Course in Algebra - Herbert E. Hawkes, William A. Luby, Frank C. Touton (1910)
• Second Course in Algebra - Herbert E. Hawkes, William A. Luby, Frank C. Touton (1911)
• Algebra: An Elementary Text-Book, Part I - G. Chrystal (1904)
• Algebra: An Elementary Text-Book, Part II - G. Chrystal (1900)
• Understanding Algebra - James W. Brennan

Abstract Algebra

• Introduction to Abstract Algebra - D. S. Malik, John N. Mordeson, M.K. Sen (Creighton University)
• Introduction to Modern Algebra - David Joyce (Clark University)
• Algebraic Methods - F. Oggier
• Abstract Algebra : Theory and Applications - Thomas W. Judson, Robert A. Beezer (Austin State University)
• An Undergraduate Course in Abstract Algebra - Robert Howlett
• Elements of Abstract and Linear Algebra - E.H. Connell (University of Miami)
• Abstract Algebra: The Basic Graduate Year - Robert Ash
• Abstract Algebra: Harvard Extension (Archived) - Benedict Gross
• Abstract Algebra: Harvard Extension Videos - Benedict Gross

Group Theory

• Notes on Group Theory - Mark Reeder
• Group Theory - J.S. Milne
• Notes on Finite Group Theory - Peter J. Cameron
• Group Theory - Pedrag Civitanovic

Linear Algebra

• Fundamentals of Linear Algebra - James B. Carrell
• Linear Algebra and Matrices - Martin Fluch
• Vector Space Theory - Robert Howlett
• Linear Algebra - Jim Hefferon
• Linear Algebra - Peter Petersen
• MIT OpenCourseWare Lectures on Linear Algebra (18.06) as Jupyter Notebooks - Juan Klopper
• Linear Algebra - Jim Hefferon
• Elementary Linear Algebra - Keith Matthews
• A First Courses in Linear Algebra - Rob Breezer
• 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)
• Linear Algebra Done Wrong - Sergei Treil
• Immersive Linear Algebra - J. Ström, K. Åström, and T. Akenine-Möller
• Interactive Linear Algebra - Dan Margalit and Joseph Rabinoff
• Linear Algebra, Infinite Dimensions, and Maple - James Herod

Ring Theory

• Foundations of Module and Ring Theory - Robert Wisbauer (University of Düsseldorf)

Galois Theory

• An Introduction to Galois Theory - Andrew Baker (University of Glasgow)
• Fields and Galois Theory - J.S. Milne
• Galois theory - Miles Reid
• Galois Theory - Ian Stewart

Lie Algebras

• Lie Algebras - Shlomo Sternberg

Combinatorics

• Basic Combinatorics - Carl G. Wagner (University of Tennessee)
• Applied Combinatorics - Mitchel T. Keller, William T. Trotter
• Notes on Combinatorics - Peter J. Cameron
• Analytic Combinatorics - Philippe Flajolet, Robert Sedgewick
• generatingfunctionology - Herbert Wilf

Graph Theory

• Graph Theory: Lecture Notes - Christopher Griffin
• Graph Theory - Reinhard Diestel

Geometry and Topology

• Fundamentals of Geometry - Oleg A. Belyaev
• A=B - M. Petkovsek, H. Wilf, D. Zeilberger
• Elements - Euclid
• Euclid's Elements Redux - Daniel Callahan
• Mathematical Illustrations - Bill Casselman
• Byrne's Euclid - Oliver Byrne
• Plane Geometry - George Wentworth and David Eugene Smith (1913)
• Planes and Spherical Trigonometry - George Wentworth and David Eugene Smith (1915)
• Coordinate Geometry - Henry Buchard Fine and Henry Dallas Thompson (1911)
• Analytic Geometry - Lewis Parker Siceloff, George Wentworth, David Eugene Smith (1922)

Differential Geometry

• Introduction to Differential Geometry - Joel W. Robbin, Dietmar A. Salamon
• Notes on Differential Geometry and Lie Groups - Jean Gallier (University of Pennsylvania)
• Topics in Differential Geometry - Peter W. Michor
• Lectures on Differential Geometry - Wulf Rossmann
• An Introduction to Riemannian Geometry - Sigmundur Gudmundsson (Lund University)
• The Geometry and Topology of Three-Manifolds - W. Thurston
• Semi-Riemann Geometry and General Relativity - Shlomo Sternberg
• Discrete Differential Geometry - Keenan Crane

Algebraic Geometry

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

Topology

• Elementary Applied Topology - Robert Ghrist (UPenn)
• Introduction to Topology
• Introduction to Topology - Alex Küronya
• Introductory Topology - Jim L. Brown
• General Topology - Pierre Schapira (Paris VI University)
• Elementary Topology Problem Textbook
• General Topology - Jesper M. Møller
• Topology Topics

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 (Paris VI University)
• Lecture Notes in Algebraic Topology - James F. Davis, Paul Kirk (Indiana University)
• Algebraic Topology - Michael Starbird
• Lecture Notes on Algebraic Topology - Jie Wu

Analysis

Real Analysis

• MIT OpenCourseWare Lectures on Calculus - G. Strang
• Elementary Calculus: An Approach Using Infinitesimals - Professor H. Jerome Keisler
• 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
• Elementary Real Analysis - Thomson, Bruckner
• Lecture Notes in Real Analysis - Eric T. Sawyer (McMaster University)
• Real Analysis - C. McMullen
• Real Analysis for Graduate Students - Richard F. Bass
• Modern Real Analysis - William P. Ziemer (Indiana University)
• Mathematical Analysis Vol I - Elias Zakon
• Mathematical Analysis Vol II - Elias Zakon
• Advanced Calculus - Lynn Loomis, Schlomo Sternberg
• 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. 1 - Wilfred Kaplan, Donald J. Lewis
• Calculus and Linear Algebra. Vol. 2 - Wilfred Kaplan, Donald J. Lewis
• Introduction to Calculus I and II - J.H. Heinbockel
• 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)
• Elements of Differential and Integral Calculus - William Anthony Granville (1911)
• Precalculus - Carl Stitz, Jeff Zeager

Harmonic Analysis

• Harmonic Analysis Lecture Notes - Richard S. Laugesen (University of Illinois at Urbana–Champaign)
• Harmonic Analysis - W. Schlag
• Lecture Notes: Fourier Transform and its Applications - Brad Osgood
• Fourier Analysis - Lucas Illing
• Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications - Julius O. Smith III (Stanford University)

Complex Analysis

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

Functional Analysis

• An Introduction to Functional Analysis - Laurent W. Marcoux (University of Waterloo)
• Functional Analysis: Lecture Notes - Jeff Schenker (Michigan State University)
• Functional Analysis Lecture Notes - T.B. Ward (University of East Anglia)
• Functional Analysis - Alexander C. R. Belton
• Topics in Real and Functional Analysis - Gerald Teschl
• Functional Analysis - Christian Remling
• Theory of Functions of a Real Variable - Shlomo Sternberg
• Functional Analysis - Lawerence Baggett

Measure Theory

• An Introduction to Measure Theory - Terence Tao (UCLA)
• Lecture Notes on Measure Theory and Functional Analysis - P. Cannarsa, T. D’Aprile
• Lecture Notes in Measure Theory - Christer Borell
• A Crash Course on the Lebesgue Integral and Measure Theory - Steve Cheng
• Measure Theory - John K. Hunter (University of California at Davis)
• Measure and Integration - Dietmar A. Salamon (ETH Zürich)
• Lecture notes: Measure Theory - Bruce K. Driver

Ordinary Differential Equations

• Difference Equations To Differential Equations - Dan Sloughter
• Ordinary Differential Equation - Alexander Grigorian (University of Bielefeld)
• Ordinary Differential Equations: Lecture Notes - Eugen J. Ionascu
• Ordinary Differential Equations - Peter Philip
• Ordinary Differential Equations - Gabriel Nagy
• Ordinary Differential Equations and Dynamical Systems - Gerald Teschl
• Notes on Differential Equations - Bob Terrell
• 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 - H. B. Phillips (1922)

Partial Differential Equations

• Notes on Partial Differential Equations - John K. Hunter (University of California at Davis)
• Partial Differential Equations: Lecture Notes - Erich Miersemann (Leipzig University)
• Linear Methods of Applied Mathematics - E. Harrell, J. Herod (Georgia Tech)

Probability and Statistics

Probability Theory

• Introduction to Probability - Charles M. Grinstead, J. Laurie Snell
• Introduction to Probability - Dimitri P. Bertsekas, John N. Tsitsiklis (MIT)
• A Short Introduction to Probability - Dirk P. Kroese (University of Queensland)
• Probability: Theory and Examples - Rick Durrett
• Probability and Statistics Cookbook - Matthias Vallentin (UC Berkeley)
• The Only Probability Cheatsheet You'll Ever Need - William Chen
• An Introduction to Probability and Random Processes - Gian-Carlo Rota, Kenneth Baclawski
• Foundations of Constructive Probability Theory - Yuen-Kwok Chan

Statistics

• Lecture Notes on Statistical Theory - Ryan Martin (University of Illinois)
• Introduction to Statistics and Data Analysis for Physicists - Gerhard Bohm, Günter Zech
• Probability and Mathematical Statistics - Prasanna Sahoo (University of Louisville)
• Lectures on Statistics - William G. Faris
• Statistical Theory - Adolfo J. Rumbos
• Theory of Statistics - James E. Gentle (George Mason University)
• Theory of Statistics - Joseph C. Watkins (University of Arizona)
• Glossary of Data Modeling - AI Access
• Statistics Papers - List of statistics papers curated by the Institute for Digital Research and Education (IDRE) at UCLA on methods such as bootstrap and factor invariance.
• NIST Handbook of Statistical Methods - Resource on practical statistics directed towards scientists and engineers.
• Concepts and Applications of Inferential Statistics - Richard Lowry
• Rough set data analysis: A road to non-invasive knowledge discovery - Ivo Düntsch, Günther Gediga
• 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

Statistical Learning

• An Introduction to Statistical Learning with Applications in R - Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani
• The Elements of Statistical Learning - Trevor Hastie, Robert Tibshirani, Jerome Friedman
• Statistical Learning Theory - Percy Liang
• Reinforcement Learning: An Introduction - Richard S. Sutton, Andrew G. Barto

Stochastic processes

• Lectures on Stochastic Processes - K. Ito (Tata Institute of Fundamental Research, Bombay)
• Probability and Stochastic Processes with Applications - Oliver Knill (Harvard University)
• Stochastic Processes - Amir Dembo (Stanford University)
• Lecture Notes on Stochastic Processes - Frank Noé, Bettina Keller and Jan-Hendrik Prinz (Freie Universität Berlin)
• Introduction to Stochastic Processes - Lecture Notes - Gordan Žitković (University of Texas)
• Applied Stochastic Processes in science and engineering - Matt Scott (University of Waterloo)
• An Introduction to Stochastic Processes in Continuous Time - Flora Spieksma (Leiden University)
• Markov Chains and Mixing Times - David A. Levin, Yuval Peres, Elizabeth L. Wilmer
• Convergence of Stochastic Processes - David Pollard

Numerical Analysis

• Introduction to Numerical Analysis - Doron Levy (University of Maryland)
• A Concise Introduction to Numerical Analysis - Douglas N. Arnold (University of Minnesota)
• Numerical Analysis - L. Ridgway Scott
• Lectures In Basic Computational Numerical Analysis - J. M. McDonough (University of Kentucky)
• Advanced Numerical Methods and Their Applications to Industrial Problems: Adaptive Finite Element Methods - Alfred Schmidt, Arsen Narimanyan
• Numerical Analysis for Engineers - Douglas Wilhelm Harder

Signal processing

• Introduction to Signal Processing - Sophocles J. Orfanidis (Rutgers University)
• Foundations of Signal Processing - Martin Vetterli, Jelena Kovacevic, Vivek K Goyal
• An Introduction to Statistical Signal Processing - Robert M. Gray, Lee D. Davisson
• Think DSP - Allen B. Downey
• Linear algebra, signal processing, and wavelets. A unified approach. - Øyvind Ryan (University of Oslo)

Mathematics for Computer Science

• Mathematics for Computer Science - Eric Lehman, F. Thomson Leighton, Albert R. Meyer
• Algorithms and Complexity - H. Wilf
• Lecture Notes on Optimization - Pravin Varaiya
• Information Theory, Inference, and Learning Algorithms - David J. C. MacKay
• The Chaos Textbook: Mathematics in the age of the computer - Glenn Elert

Mathematical Biology

• Mathematical Biology - Jeffrey Chasnov

Mathematical Physics

• Introduction to Continuum Mechanics - Ray. M. Bowen
• Mathematical Tools for Physics - James Nearing
• Mechanism of the Heavens (1831) - Mary Somerville

Students Lecture 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.

License

To the extent possible under law, Cyrille Rossant has waived all copyright and related or neighboring rights to this work.