Awesome maths

A curated list of awesome mathematics resources.

Foundations of mathematics

Set theory

• 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

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

Homotopy type theory

• Homotopy type theory

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

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)

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

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

Group theory

• Notes on Group Theory - Mark Reeder
• Group theory - J.S. Milne
• Notes on finite group theory - Peter J. Cameron

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

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

Analysis

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 - Jirí Lebl
• Elementary Real Analysis - Thomson, Bruckner
• Lecture Notes in Real Analysis - Eric T. Sawyer (McMaster University)
• Real analysis - C. McMullen
• From Calculus to Analysis - Steen Pedersen
• Real analysis for graduate students - Richard F. Bass
• Modern Real Analysis - William P. Ziemer (Indiana University)

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

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

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

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

• 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

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)

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

Graph theory

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

Geometry and topology

• Fundamentals of Geometry - Oleg A. Belyaev

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)

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)

Topology

• 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

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

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)

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 - http://mason.gmu.edu/~jgentle/books/MathStat.pdf
• Theory of Statistics - Joseph C. Watkins (University of Arizona)

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

Numerical Analysis

Computer algebra

Websites

Learning platforms

• Khan Academy
• Coursera
• Wikibooks

Questions and answers

• Mathematics Stack Exchange
• MathOverflow - for professional mathematicians

Encyclopedia

• Encyclopedia of Mathematics
• Wolfram Mathworld
• Planetmath

Misc

• Areas of mathematics on Wikipedia

Books

• Basics of Algebra, Topology, and Differential Calculus - Jean Gallier (University of Pennsylvania)

Online courses and MOOCs

Videos