Geometry

Differential Geometry

• Introduction to Tensor Calculus and Continuum Mechanics, by John H. Heinbockel
• Topics in Differential Geometry, Peter W. Michor.
• Natural operations in differential geometry by Ivan Kolar, Jan Slovak and Peter W. Michor
• Intrinsic Geometry of Surfaces by A. D. Aleksandrov and V. A. Zalgaller
• An Introduction to Riemannian Geometry - Sigmundur Gudmundsson, Lund University
• Riemann Surfaces, Dynamics and Geometry Curtis McMullen
• Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem by Peter B. Gilkey
• Differentiable Manifolds lecture notes (growing) by Mariusz Wodzicki
• Lecture notes for the course in Differential Geometry by Sergei Yakovenko
• Lectures On Symplectic Geometry with and Eye Toward Combinatorics by Sue Tolman
• Differential Geometry lecture notes by Gabriel Lugo
• Differential geometry reconstructed by Alan U. Kennington
• Course of Differential Geometry by Ruslan Sharipov
• A Modern Course on Curves and Surfaces by Richard S. Palais
• Undergraduate notes on Projective Geometry, Manifolds etc. by Nigel Hitchin
• Vector Bundles and an Introduction to Gauge Theory by Steven Bradlow, (Hung-Jen Hsu and Chris Willett)
• Twistor Geometry and Gauge Theory by Martin Wolf
• Complex Geometry of Nature and General Relativity by Giampiero Esposito

• Geometric Asymptotics by Victor Guillemin and Shlomo Sternberg
• Projective Geometry by Balázs Csikós
• Differential Geometry by Balázs Csikós
• Projective and Polar Spaceas by Peter Cameron
• Transformation groups for begginers by S. V. Duzhin and B. D. Tcherbotarevsky

Cartan's Corner Differential geometry & Topology

Analysis

• Homeomorphisms in Analysis by Casper Goffman, Togo Nishiura, and Daniel Waterman
• Why the Boundary of a Round Drop Becomes a Curve of Order Four by A. N. Varchenko and P. I. Etingof
• Lectures on Entire Functions by B. Ya. Levin
• Constructive Real Numbers and Constructive Function Spaces by N. A. Sanin
• Complex Variables by Robert B. Ash and W.P. Novinger
• Foundations of Nonstandard Analysis by Erland Gadde
• Analytic functions by Stanislaw Saks, Antoni Zygmund
• Theory of Functions of real variable by Shlomo Sternberg

Functional Analysis

• Functional Analysis by Douglas N. Arnold
• Paul Garrett's lecture notes
• Functional Analysis (3rd year) lecture notes by Thomas Ward
• Functional Analysis and Semi-Groups by Einar Hille and R. S. Phillips, AMS 1957, 808 pp
• The Convenient Setting of Global Analysis by Andreas Kriegl and Peter W. Michor, AMS 1997, 618pp
• Hilbert Space Methods for Partial Differential Equations, by R.E. Showalter
• Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, by R. E. Showalter
• Harmonic Analysis and Partial Differential Equations - Björn E.J. Dahlberg and Carlos E. Kenig
• Functional Analysis by Thomas Ward
• Harmonic Function Theory by Sheldon Axler, Paul Bourdon, and Wade Ramey printing disabled
• An Introduction to C*-Algebras by Pierre de la Harpe and Vaughan Jones
• Functional Analysis by Dr. P. G. Dixon
• Introduction to Microlocal Analysis by Richard Melrose

• Bernoulli Periodic distributions Notes by Miguel Lerma
• Holomorphic Methods in Analysis and Mathematical Physics by Brian C. Hall

Differential Equations

• Nonlinear Analysis and Ordinary Differential Equations by Klaus Schmitt
• Ordinary Differential Equations and Dynamical Systems by Gerald Teschl
• Ordinary Differential Equations Notes by Sheldon Newhouse
• Partial Differential Equations by William W. Symes
• Lectures on analytic differential equations by Yulij Ilyashenko and Sergei Yakovenko
• Exterior Differential Systems and Euler-Lagrange Partial Differential Equations by Robert L. Bryant, Phillip A. Griffiths, Daniel A. Grossman
• Monograph on quasilinear partial differential equations by Marco Squassina

Algebra, Algebraic Geometry

Algebra, Abstract

• Lectures on Matrices by J. H. M. Wedderburn
• Elementary Abstract Algebra by Edwin Clark
• Intro to Abstract Algebra by Paul Garrett
• Elements of Abstract and Linear Algebra by Edwin H. Connell
• Abstract Algebra:The Basic Graduate Year by Robert Ash
• A Course In Commutative Algebra by Robert Ash
• Notes for geometric group theory course, by John R. Stallings
• Advanced Linear Algebra lecture notes by Keith Matthews (with real algorithm for the real Jordan form)
• Commutative algebra by Ivan Fesenko
• Abstract Algebra - Number Theory, Group Theory, Galois Theory by Daviiddd Wilkins
• Workbook in Higher Algebra by David Surowski with index, many exercises
• Abstract group theory Papers and Notes by Chuck Miller
• The Classification of the Finite Simple Groups by Daniel Gorenstein, Richard Lyons, and Ronald Solomon
• Group Characters, Symmetric Functions, and the Hecke Algebra by David M. Goldschmidt
• Finite Groups and their Representations by Andrew Baker
• Classical Groups by Peter Cameron
• Galois theory by Miles Reid
• Toeplitz and Circulant Matrices by R. M. Gray
• Homogeneous spaces and equivariant embeddings by Dmitri A. Timashev

• Differential Algebra by Joseph Fels Ritt
• Structure and Representations of Jordan Algebras by Nathan Jacobson
• An Invitation to General Algebra and Universal Constructions, by George M. Bergman
• A Course in Universal Algebra by Stanley N. Burris and H.P. Sankappanavar
• Fundamental Problems in Algorithmic Algebra by Chee Yap
• Braid groups and Galois theory by Frazer Jarvis
• A. N. Whitehead's Universal Algebra by Stephen Blake
• Varieties of Lattices by Peter Jipsen and Henry Rose
• Commutator Theory for Congruence Modular Varieties by Ralph Freese and Ralph McKenzie
• Constructive Aspects of the Inverse Galois Problem by C. Jensen, A. Ledet, and N. Yui
• Publications and prepublications about braids

Algebraic Geometry

• Algebraic geometry and projective differential geometry by Joseph M. Landsberg
• Introduction to Algebraic Geometry, Igor V. Dolgachev
• Notes on Algebraic Geometry, J.S. Milne.
• Introduction to Algebraic Geometry by Yuriy Drozd
• Undergraduate tutorial on modular forms by Marty Weissman
• Modular forms (Igor Dolgachev)
• Complex analytic and algebraic geometry - Jean-Pierre Demailly
• Moduli Spaces in Algebraic Geometry edited by L. Goettsche
• Algebraic Curves by Hendrik L. Lenstra
• Lectures by David A. Cox - Introduction to Algebraic Geometry, Minicourse on Toric Varieties, etc.
• Quasi-projective Moduli for Polarized Manifolds by Eckart Viehweg
• Lectures on Vanishing Theorems by Hélène Esnault, Eckart Viehweg
• Introduction to Algebraic Geometry by Ravi Vakil
• Determinantal rings by Winfried Bruns and Udo Vetter
• Simplification of Hironaka's resolution of singularities by János Kollár
• Eleven books on geometry, topology, and algebra by Andrew Ranicki

Elliptic Curves

• Elliptic curves by Miles Reid
• Elliptic curves, student project, Annegret Weng (ps 726K)
• Arithmetic on hyperelliptic curves, Annegret Weng (ps 148K)
• Course Notes by Jim Milne: Elliptic Curves, Modular functions and forms, Abelian varieties, Etale Cohomology
• The Modular curves X₀(N), Lecture notes by Bas Edixhoven
• Expository articles - Computing rational points on curves, Elliptic curves by Bjorn Poonen

Category Theory

• A Gentle Introduction to Category Theory, by Maarten M.Fokkinga
• Mixed Motives by Marc Levine
• Chris Hillman's Eprints has a concise introduction to category theory.
• David Madore's Math Page has a 50 page treatise on categories, in DVI and PS.
• Lecture notes on Categories, Derived Categories, Homological Algebra by Pierre Schapira
• Toposes, Triples and Theories by Michael Barr and Charles Wells
• Higher-Dimensional Categories: an illustrated guide book by Eugenia Cheng and Aaron Lauda
• Abstract and Concrete Categories by J. Adamek, H. Herrlich, G. Strecker
• Higher Operads, Higher Categories by Tom Leinster

Lie Algebras

• Algebraic Groups and Discontinuous Subgroups by Armand Borel and George D. Mostow
• Theory of representations etc. by Claudio Procesi
• Semisimple Lie Algebras and their Representations, by Robert N. Cahn,
• Linear Algebraic Groups - minor thesis by David Dumas
• Lie Algebras by Shlomo Sternberg

Number Theory

Introductory Number Theory

• An Introduction to the Theory of Numbers by Leo Moser
• Elementary Number Theory by W. Edwin Clark
• Lecture notes on elementary number theory (Bruce Ikenaga)
• exercizes with solutions on elementary number theory (Nigel Byott)
• Naoki Sato's Number Theory Notes
• Math 780: Elementary Number Theory, Notes by Michael Filaseta, 1997

p-adic numbers

• Lecture notes on p-adic numbers (Andrew Baker)

Arithmetic Geometry

• Draft of PCMI Lecture Notes on Open Questions in Arithmetic Algebraic Geometry (Alice Silverberg)

Fermat's Last Theorem

• William Hammond's Fermat Archive
• Notes on Fermat's Last Theorem, A.J. van der Poorten, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley-Interscience, January, 1996, ISBN 0-471-06261-8
  • Review of Alf van der Poorten's Notes on Fermat's Last Theorem by Andrew Granville

Algebraic Number Theory

• A Computational Introduction to Number Theory and Algebra by Victor Shoup
• A Course In Algebraic Number Theory by Robert B. Ash
• Course Notes for elementary and algebraic number theory by Ivan Fesenko
• Course Notes by Jim Milne: Algebraic Number Theory
• Algebraic Number Theory, Course notes by Robin Chapman
• Invitation to higher local fields, by Ivan Fesenko and Masato Kurihara (editors)
• Survey of Euclidean Number Fields (ps file 371K) (Franz Lemmermeyer)
• Algebra and Number Theory by Andrew Baker
• Elementary Number Theory by William A. Stein
• Notes for Dick Gross' Algebraic Number Theory Course by S. Sather-Wagstaff
• Algebraic Number Theory & Introduction to Automorphic L-functions by Ching-Li Chai
• Lecture notes on Algebraic Number Theory by Jerome Hoffmann
• Introduction to Number Theory by Franz Lemmermeyer

Analytic Number Theory

• Distribution of Prime Numbers, Lecture notes by William Chen
• Course notes for analytic number theory (Graham Everest) (ps 442K)
• Introduction to Analytic Number Theory by Noam Elkies

General Mathematics

elementary mathematics, Analysis, Lebesgue Integrals ..

• Basic Concepts of Mathematics by Elias Zakon
• Mathematical Analysis I by Elias Zakon
• Calculus by Gilbert Strang, also Student Study Guide to Calculus and Instructor's Manual
• Basic algebra and trigonometry. by Thomas Ward
• First Year Mathematics, (notes by William Chen)
• Multivariable Calculus by George Cain & James Herod
• Complex Analysis by George Cain
• Linear Algebra byJim Hefferon
• Elementary Linear Algebra by Keith Matthews
• A First Course in Linear Algebra by Robert A. Beezer (GNU Free Documentation License)
• Measure Theory and Integration by V. Liskevich
• MEASURE THEORY by D.H.Fremlin plain tex sources
• Complex Analysis by Douglas N. Arnold
• Real Analysis for Engineers by Erhan Cinlar and Robert J. Vanderbei
• Set Theoretic Real Analysis by Krzysztof Ciesielski
• Expository notes by Daniel Allcock in The finite reflection groups
• Classical Geometry by Danny Calegari
• Classical Geometry and Low-Dimensional Topology by Danny Calegari
• Difference Equations to Differential Equations by Dan Sloughter
• Course of linear algebra and multidimensional geometry by Ruslan Sharipov
• Examples, Lecture Notes and Specimen Exam Questions and Natural Sciences Tripos
• Calculus project of Michael Livshits
• The Matrix Cookbook by Kaare Brandt Petersen
• The Octonions by John C. Baez
• Advanced Calculus by Shlomo Sternberg

Mathematical Logic and Set theory

• Mathematical Logic - an Introduction by Michal Walicki
• Set theory by Dr. P. G. Dixon
• Mathematical Logic by Stephen G. Simpson
• Foundations of Mathematics by Stephen G. Simpson
• Model Theory by Stephen G. Simpson
• The theory of the foundations of mathematics - 1870 to 1940
• A Problem Course in Mathematical Logic by Stefan Bilaniuk
• Computability and Incompleteness by Jeremy Avigad
• Complete proofs of Gödel's Incompleteness Theorems
• Fundamentals of Model Theory by William Weiss and Cherie D'Mello
• Logic for Everyone by Robert A. Herrmann
  For more logic lecture notes go to: ASSOCIATION FOR SYMBOLIC LOGIC members.
  For more set theory materials go to: Set Theory Preprint Sites

Discrete Math

• Lecture Notes on Discrete Mathematics by László Lovász