# Awesome maths

A curated list of awesome mathematics resources.


## Foundations of mathematics

### Set theory

* [An introduction to set theory](http://www.math.toronto.edu/weiss/set_theory.pdf) - William A. R. Weiss
* [Set Theory and Foundations of Mathematics](http://www.settheory.net/) - Sylvain Poirier
* [Set Theory on the Stanford Encyclopedia of Philosophy](http://plato.stanford.edu/entries/set-theory/)

### Logic

* [Introduction to logic](http://logic.stanford.edu/intrologic/chapters/cover.html) - Michael Genesereth, Eric Kao (Stanford University)
* [An introduction to formal logic](https://textbookequity.org/oct/Textbooks/Magnus_forallx.pdf) - P.D. Magnus (University at Albany)
* [A problem course in mathematical logic](http://euclid.trentu.ca/math/sb/pcml/pcml-16.pdf) - Stefan Bilaniuk (Trent University)
* [Language, proof, and logic](http://ssdi.di.fct.unl.pt/~pb/cadeiras/lc/0102/lpl%20textbook.pdf) - Jon Barwise, John Etchemendy
* [Mathematical logic](http://www.mathematik.uni-muenchen.de/~schwicht/lectures/logic/ws03/ml.pdf) - Helmut Schwichtenberg
* [Mathematical logic](http://www.personal.psu.edu/t20/notes/logic.pdf) - Stephen G. Simpson (Pennsylvania State University)
* [Formal logic](http://maude.sip.ucm.es/~miguelpt/papers/flogic.pdf) - Miguel Palomino

### Category theory

* [Introduction to category theory and categorical logic](http://www.mathematik.tu-darmstadt.de/~streicher/CTCL.pdf) - Thomas Streicher
* [An introduction to Category Theory](http://www.cs.man.ac.uk/~hsimmons/zCATS.pdf) - Harold Simmons
* [Category Theory](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.211.4754&rep=rep1&type=pdf) - Steve Awodey (Carnegie Mellon University)
* [Category Theory](http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/04SS/Cats1.pdf) - B. Pareigis
* [Category Theory for Computing Science](http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf) - Michael Barr, Charles Wells

### Homotopy type theory

* [Homotopy type theory](https://hottheory.files.wordpress.com/2013/03/hott-online-611-ga1a258c.pdf)


## Number theory

* [Elementary Number Theory: Primes, Congruences, and Secrets](http://wstein.org/ent/ent.pdf) - William Stein
* [Elementary Number Theory](http://www.iiserpune.ac.in/~ayan/MTH312/%28127%29.pdf) - W. Edwin Clark (University of South Florida)
* [A Course on Number Theory](http://www.maths.qmul.ac.uk/~pjc/notes/nt.pdf) - Peter J. Cameron
* [A Computational Introduction to Number Theory and Algebra](http://shoup.net/ntb/ntb-v2.pdf) - Victor Shoup
* [Number Theory: A Contemporary Introduction](http://math.uga.edu/~pete/4400FULL.pdf) - Pete L. Clark

### Algebraic number theory

* [Introduction to Algebraic Number Theory](http://www1.spms.ntu.edu.sg/~frederique/ANT10.pdf)
* [Introduction to Algebraic Number Theory](http://www1.spms.ntu.edu.sg/~frederique/ANT10.pdf) - F. Oggier
* [Algebraic Number Theory](http://www.jmilne.org/math/CourseNotes/ANT.pdf) - J.S. Milne
* [Algebraic Number Theory Course Notes](http://people.math.gatech.edu/~mbaker/pdf/ANTBook.pdf) - Matthew Baker (Georgia Tech)

### Analytic number theory

* [Introduction to Analytic Number Theory](http://www.math.uiuc.edu/~hildebr/ant/main.pdf) - A.J. Hildebrand (University of Illinois)
* [Elements of analytic number theory](http://math.nsc.ru/~vdovin/lectures/numth_eng.pdf) - P. S. Kolesnikov, E. P. Vdovin (Novosibirsk)
* [Analytic Number Theory](http://www.mathematik.uni-muenchen.de/~forster/v/ann/annth_all.pdf) - Otto Forster (LMU Munich)
* [Analytic number theory - Lecture notes based on Davenport’s book](http://www2.math.uu.se/~astrombe/analtalt08/www_notes.pdf) - Andreas Strömbergsson


## Algebra

### Abstract algebra

* [Introduction to Abstract Algebra](https://people.creighton.edu/~dsm33733/MTH581/Introduction%20to%20Abstract%20Algebra.pdf) - D. S. Malik, John N. Mordeson, M.K. Sen (Creighton University)
* [Introduction to Modern Algebra](http://aleph0.clarku.edu/~djoyce/ma225/algebra.pdf) - David Joyce (Clark University)
* [Algebraic methods](http://www1.spms.ntu.edu.sg/~frederique/AA11.pdf) - F. Oggier
* [Abstract Algebra : Theory and Applications](http://abstract.ups.edu/download/aata-20150812.pdf) -  Thomas W. Judson, Robert A. Beezer (Austin State University)
* [An undergraduate course in Abstract Algebra](http://www.maths.usyd.edu.au/u/bobh/UoS/rfwhole.pdf) - Robert Howlett

### Group theory

* [Notes on Group Theory](https://www2.bc.edu/~reederma/Groups.pdf) - Mark Reeder
* [Group theory](http://www.jmilne.org/math/CourseNotes/GT.pdf) - J.S. Milne
* [Notes on finite group theory](http://www.maths.qmul.ac.uk/~pjc/notes/gt.pdf) - Peter J. Cameron

### Linear algebra

* [Fundamentals of linear algebra](http://www.math.ubc.ca/~carrell/NB.pdf) - James B. Carrell
* [Linear algebra and matrices](http://mathstat.helsinki.fi/~fluch/linear_algebra_1-sp07/la1.pdf) - Martin Fluch
* [Vector space theory](http://www.maths.usyd.edu.au/u/bobh/UoS/MATH2902/vswhole.pdf) - Robert Howlett
* [Linear Algebra](http://www.mathematik.uni-muenchen.de/~renesse/AP09/hefferon.pdf) - Jim Hefferon
* [Linear algebra](http://www.calpoly.edu/~jborzell/Courses/Year%2010-11/Fall%202010/Petersen-Linear_Algebra-Math_306.pdf) - Peter Petersen

### Ring theory

* [Foundations of Module and Ring Theory](http://www.math.uni-duesseldorf.de/~wisbauer/book.pdf) - Robert Wisbauer (University of Düsseldorf)

### Galois theory

* [An Introduction to Galois Theory](http://www.maths.gla.ac.uk/~ajb/dvi-ps/Galois.pdf) - Andrew Baker (University of Glasgow)
* [Fields and Galois Theory](http://www.jmilne.org/math/CourseNotes/FT.pdf) - J.S. Milne
* [ Galois theory](http://homepages.warwick.ac.uk/~masda/MA3D5/Galois.pdf) - Miles Reid


## Analysis

### Real analysis

* [An Introduction to Real Analysis](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.pdf) - John K. Hunter (University of California at Davis)
* [Introduction to real analysis](http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF) - William F. Trench (Trinity University, Texas)
* [Basic Analysis: Introduction to Real Analysis](http://www.jirka.org/ra/realanal.pdf) - Jirí Lebl
* [Elementary Real Analysis](http://prac.im.pwr.wroc.pl/~kwasnicki/pl/stuff/tbb-hyper.pdf) - Thomson, Bruckner
* [Lecture Notes in Real Analysis](http://ms.mcmaster.ca/~sawyer/Publications/Real_Analysis.pdf) - Eric T. Sawyer (McMaster University)
* [Real analysis](http://math.harvard.edu/~ctm/papers/home/text/class/harvard/212a/course/course.pdf) - C. McMullen
* [From Calculus to Analysis](http://www.wright.edu/~steen.pedersen/teaching/Calculus2Analysis.pdf) - Steen Pedersen
* [Real analysis for graduate students](http://homepages.uconn.edu/~rib02005/rags010213.pdf) - Richard F. Bass
* [Modern Real Analysis](http://www.indiana.edu/~mathwz/PRbook.pdf) - William P. Ziemer (Indiana University)

### Harmonic analysis

* [Harmonic Analysis Lecture Notes](http://www.math.uiuc.edu/~laugesen/545/545Lectures.pdf) - Richard S. Laugesen (University of Illinois at Urbana–Champaign)
* [Harmonic Analysis](http://www.math.uchicago.edu/~schlag/harmonicnotes.pdf) - W. Schlag
* [Lecture Notes: Fourier Transform and its Applications](https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf) - Brad Osgood
* [Fourier Analysis](http://www.reed.edu/physics/courses/Physics331.f08/pdf/Fourier.pdf) - Lucas Illing

### Complex analysis

* [Introduction to Complex Analysis](http://www.unc.edu/math/Faculty/met/complex.pdf) - Michael Taylor
* [An Introduction to Complex Analysis and Geometry](http://www.math.uiuc.edu/~jpda/jpd-complex-geometry-book-5-refs-bip.pdf) - John P. D'Angelo (University of Illinois)
* [A First Course in Complex Analysis](http://math.sfsu.edu/beck/papers/complex.pdf) - Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka
* [A Guide to Complex Variables](http://www.math.wustl.edu/~sk/books/guide.pdf) - Steven G. Krantz
* [Complex Analysis](http://www.maths.manchester.ac.uk/~cwalkden/complex-analysis/complex_analysis.pdf) - Charles Walkden
* [Complex Analysis](http://www.math.ku.dk/noter/filer/koman-12.pdf) - Christian Berg
* [Complex Variables](http://people.math.sc.edu/girardi/m7034/book/AshComplexVariablesWithHyperlinks.pdf) - R. B. Ash, W.P. Novinger
* [Complex Analysis](http://www.maths.lth.se/matematiklu/personal/olofsson/CompHT06.pdf) - Christer Bennewitz
* [Complex Analysis](https://www.math.washington.edu/~marshall/math_536/Notes.pdf) - Donald E. Marshall
* [A concise course in complex analysis and Riemann surfaces](http://www.math.uchicago.edu/~schlag/bookweb.pdf) - Wilhelm Schlag

### Functional analysis

* [An Introduction to Functional Analysis](http://www.math.uwaterloo.ca/~lwmarcou/Preprints/LinearAnalysis.pdf) - Laurent W. Marcoux (University of Waterloo)
* [Functional Analysis: Lecture Notes](http://users.math.msu.edu/users/jeffrey/920/920notes.pdf) - Jeff Schenker (Michigan State University)
* [Functional analysis lecture notes](http://www.math.hkbu.edu.hk/~zeng/Teaching/math3680/FAnotes.pdf) - T.B. Ward (University of East Anglia)
* [Functional analysis](http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf) - Alexander C. R. Belton
* [Topics in Real and Functional Analysis](https://www.mat.univie.ac.at/~gerald/ftp/book-fa/fa.pdf) - Gerald Teschl
* [Functional Analysis](http://www2.math.ou.edu/~cremling/teaching/lecturenotes/fa-new/LN-I.pdf) - Christian Remling

### Measure theory

* [An introduction to measure theory](https://terrytao.files.wordpress.com/2011/01/measure-book1.pdf) - Terence Tao (UCLA)
* [Lecture Notes on Measure Theory and Functional Analysis](http://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf) - P. Cannarsa, T. D’Aprile
* [Lecture Notes in Measure Theory](http://www.math.chalmers.se/~borell/MeasureTheory.pdf) - Christer Borell
* [A Crash Course on the Lebesgue Integral and Measure Theory](http://www.gold-saucer.org/math/lebesgue/lebesgue.pdf) - Steve Cheng
* [Measure Theory](https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes.pdf) - John K. Hunter (University of California at Davis)
* [Measure and Integration](https://people.math.ethz.ch/~salamon/PREPRINTS/measure.pdf) - Dietmar A. Salamon (ETH Zürich)
* [Lecture notes: Measure Theory](http://www.math.ucsd.edu/~bdriver/240-00-01/Lecture_Notes/measurep.pdf) - Bruce K. Driver

### Ordinary Differential Equations

* [Ordinary Differential Equation](https://www.math.uni-bielefeld.de/~grigor/odelec2008.pdf) - Alexander Grigorian (University of Bielefeld)
* [Ordinary Differential Equations: Lecture Notes](http://www.cs.bgu.ac.il/~leonid/ode_bio_files/Ionascu_LectNotes.pdf) - Eugen J. Ionascu
* [Ordinary Differential Equations](http://www.math.lmu.de/~philip/publications/lectureNotes/ODE.pdf) - Peter Philip
* [Ordinary Differential Equations](http://users.math.msu.edu/users/gnagy/teaching/ode.pdf) - Gabriel Nagy
* [Ordinary Differential Equations and Dynamical Systems](http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf) - Gerald Teschl

### Partial differential equations

* [Notes on Partial Differential Equations](https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf) - John K. Hunter (University of California at Davis)
* [Partial Differential Equations: Lecture Notes](http://www.math.uni-leipzig.de/~miersemann/pdebook.pdf) - Erich Miersemann (Leipzig University)


## Combinatorics

* [Basic Combinatorics](http://www.math.utk.edu/~wagner/papers/comb.pdf) - Carl G. Wagner (University of Tennessee)
* [Applied Combinatorics](http://people.math.gatech.edu/~trotter/book.pdf) - Mitchel T. Keller, William T. Trotter
* [Notes on Combinatorics](http://www.maths.qmul.ac.uk/~pjc/notes/comb.pdf) - Peter J. Cameron
* [Analytic Combinatorics](http://algo.inria.fr/flajolet/Publications/book.pdf) - Philippe Flajolet, Robert Sedgewick

### Graph theory

* [Graph Theory: Lecture Notes](http://www.personal.psu.edu/cxg286/Math485.pdf) - Christopher Griffin
* [Graph Theory](http://www.cs.unibo.it/babaoglu/courses/cas00-01/tutorials/GraphTheory.pdf) - Reinhard Diestel


## Geometry and topology

* [Fundamentals of Geometry](http://polly.phys.msu.ru/~belyaev/geometry.pdf) - Oleg A. Belyaev

### Differential geometry

* [Introduction to differential geometry](https://people.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf) - Joel W. Robbin, Dietmar A. Salamon
* [Notes on Differential Geometry and Lie Groups](http://www.seas.upenn.edu/~jean/diffgeom.pdf) - Jean Gallier (University of Pennsylvania)
* [Topics in Differential Geometry](http://www.mat.univie.ac.at/~michor/dgbook.pdf) - Peter W. Michor
* [Lectures on Differential Geometry](http://mysite.science.uottawa.ca/rossmann/Differential%20Geometry%20book_files/Diffgeo.pdf) - Wulf Rossmann
* [An Introduction to Riemannian Geometry](http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf) - Sigmundur Gudmundsson (Lund University)

### Algebraic geometry

* [A Brief Introduction to Algebraic Geometry](http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/rickksda1107.pdf) - R.C. Churchill
* [Introduction to Algebraic Geometry](http://www.math.lsa.umich.edu/~idolga/631.pdf) - Igor V. Dolgachev
* [Foundations of Algebraic Geometry](http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf) - Ravi Vakil
* [Algebraic Geometry](http://www.cis.upenn.edu/~jean/algeoms.pdf) - Jean Gallier, Stephen S. Shatz (University of Pennsylvania)
* [Algebraic Geometry](http://www.jmilne.org/math/CourseNotes/AG.pdf) - J.S. Milne
* [Algebraic Geometry](http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf) - Andreas Gathmann (University of Kaiserslautern)

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### Algebraic topology

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## Probability and statistics

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* [Statistical Learning Theory](https://web.stanford.edu/class/cs229t/notes.pdf) - Percy Liang

## Numerical Analysis

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## Computer algebra

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# Websites

## Learning platforms

* [Khan Academy](https://www.khanacademy.org/math)
* [Coursera](https://www.coursera.org/courses?query=mathematics&languages=en)
* [Wikibooks](https://en.wikibooks.org/wiki/Wikibooks:Mathematics_bookshelf)

## Questions and answers

* [Mathematics Stack Exchange](http://math.stackexchange.com/)
* [MathOverflow](http://mathoverflow.net/) - for professional mathematicians

## Encyclopedia

* [Encyclopedia of Mathematics](https://www.encyclopediaofmath.org)
* [Wolfram Mathworld](http://mathworld.wolfram.com/)
* [Planetmath](http://planetmath.org/)

## Misc

* [Areas of mathematics on Wikipedia](https://en.wikipedia.org/wiki/Areas_of_mathematics)



# Books

* [Basics of Algebra, Topology, and Differential Calculus](http://www.cis.upenn.edu/~jean/math-basics.pdf) - Jean Gallier (University of Pennsylvania)


# Online courses and MOOCs



# Videos