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Let me start by apologizing if there is another thread on math.se that subsumes this.

I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting through books to find [the best source]". It strikes me now that while I love books (I really do) I often find that I learn best from sets of lecture notes and short articles. There are three particular reasons that make me feel this way.

1.Lecture notes and articles often times take on a very delightful informal approach. They generally take time to bring to the reader's attention some interesting side fact that would normally be left out of a standard textbook (lest it be too big). Lecture notes and articles are where one generally picks up on historical context, overarching themes (the "birds eye view"), and neat interrelations between subjects.

2.It is the informality that often allows writers of lecture notes or expository articles to mention some "trivial fact" that every textbook leaves out. Whenever I have one of those moments where a definition just doesn't make sense, or a theorem just doesn't seem right it's invariably a set of lecture notes that sets everything straight for me. People tend to be more honest in lecture notes, to admit that a certain definition or idea confused them when they first learned it, and to take the time to help you understand what finally enabled them to make the jump.

3.Often times books are very outdated. It takes a long time to write a book, to polish it to the point where it is ready for publication. Notes often times are closer to the heart of research, closer to how things are learned in the modern sense.

It is because of reasons like this that I find myself more and more carrying around a big thick manila folder full of stapled together articles and why I keep making trips to Staples to get the latest set of notes bound.

So, if anyone knows of any set of lecture notes, or any expository articles that fit the above criteria, please do share!

I'll start:

People/Places who have a huge array of fantastic notes:

  1. K Conrad

  2. Pete L Clark

  3. Milne

  4. Stein

  5. Igusa

  6. Hatcher

  7. Andrew Baker (Contributed by Andrew)

  8. Garrett (Contributed by Andrew)

  9. Frederique (Contributed by Mohan)

  10. Ash

  11. B Conrad

  12. Matthew Emerton (not technically notes, but easily one of the best reads out there).

  13. Geraschenko

  14. A collection of the "What is..." articles in the Notices

  15. Brian Osserman

Model Theory:

  1. The Model Theory of Fields-Marker

Number Theory:

  1. Algebraic Number Theory-Conrad
  2. Algebraic Number Theory-Weston
  3. Class Field Theory-Lemmermeyer
  4. Compilation of Notes from Things of Interest to Number Theorists
  5. Elliptic Modular Forms-Don Zagier
  6. Modular Forms-Martin
  7. What is a Reciprocity Law?-Wyman
  8. Class Field Theory Summarized-Garbanati
  9. Three Lectures About the Arithmetic of Elliptic Curves-Mazur
  10. Congruences Between Modular Forms-Calegari
  11. Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture-Rubin
  12. Simple Proof of Kronecker Weber-Ordulu
  13. Tate's Thesis-Binder
  14. Introduction to Tate's Thesis-Leahy
  15. A Summary of CM Theory of Elliptic Curves-Getz
  16. An Elementary Introduction to the Langland's Program-Gelbart
  17. $p$-adic Analysis Compared to Real Analysis-Katok (Contributed by Andrew)
  18. Representation of $p$-adic Groups-Vinroot
  19. Counting Special Points: Logic, Diophantine Geometry, and Transcendence Theory-Scanlon
  20. Algebraic Number Theory-Holden

Complex Geometry:

  1. Complex Analytic and Differential Geometry-Demailly

  2. Weighted $L^2$ Estimes for the $\bar{\partial}$ Operator on a Complex Manifold Demailly

  3. Uniformization Theorem-Chan

  4. Analytic Vector Bundles-Andrew (These notes are truly amazing)

  5. Complex Manifolds-Koppensteiner

Differential Topology/Geometry:

  1. Differential Topology-Dundas

  2. Spaces and Questions-Gromov

  3. Introduction to Cobordism-Weston

  4. The Local Structure of Smooth Maps of Manifolds-Bloom

  5. Groups Acting on the Circle-Ghys

  6. Lie Groups-Ban (comes with accompanying lecture videos)

  7. Very Basic Lie Theory-Howe

  8. Differential Geometry of Curves and Surfaces-Shifrin (Contributed by Andrew)

  9. A Visual Introduction to Riemannian Curvatures and Some Discrete Generlizations-Ollivier

Algebra:

  1. Geometric Group Theory-Bowditch

  2. Categories and Homological Algebra-Schapira

  3. Category Theory-Leinster (Contributed by Bruno Stonek)

  4. Category Theory-Chen (Contributed by Bruno Stonek)

  5. Commutative Algebra-Altman and Klein (Contributed by Andrew)

  6. Finite Group Representation Theory-Bartel (Contributed by Mohan)

  7. Representation Theory-Etingof

  8. Commutative Algebra-Haines

  9. Geometric Commutative Algebra-Arrondo

  10. Examples in Category Theory-Calugereanu and Purdea

Topology

  1. Homotopy Theories and Model Categories-Dwyer and Spalinski (Contributed by Elden Elmanto)

Algebraic Geometry:

  1. Foundations of Algebraic Geometry-Vakil

  2. Analytic Techniques in Algebraic Geometry-Demailly

  3. Algebraic Geometry-Gathmann (Contributed by Mohan)

  4. Oda and Mumford's Algebraic Geometry Notes (Pt. II)

  5. Galois Theory for Schemes-Lenstra

  6. Rational Points on Varieties-Poonen

NOTE: This may come in handy for those who, like me, don't like a metric ton of PDFs associated to a single document: http://www.pdfmerge.com/

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5  
Wouldn't this fit better on a wiki or so? –  vonbrand Feb 13 '13 at 12:36
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here are some great ones rutherglen.science.mq.edu.au/wchen/ln.html –  user58512 Feb 13 '13 at 21:13
    
@user58512 Good one! You should put it as an answer! –  Alex Youcis Feb 13 '13 at 21:14
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It's a bit sad to admit, but seeing this thread for the first time just earlier tonight I feared that somewhere in its history it was tagged [set-theory]... I was relieved to see that it wasn't the case! :-) –  Asaf Karagila Mar 8 '13 at 1:42
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@darijgrinberg Oh--you're not on a first-name basis with Frederique? –  Alex Youcis Mar 28 '13 at 5:31
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6 Answers

In no particular order:

If we're going to mention Hatcher (famous to me for the algebraic topology notes), we might as well also mention a few other books that are online, like Algebra chapter 0, Stanley's insane first volume of Enumerative Combinatorics (which reminds me: generatingfunctionology). Also I don't see topology without tears mentioned. The sheer number of books and notes on differential geometry and lie theory is mind-boggling, so I'll have to update later with the juicier ones.

Let's not forget the AMS notes online back through 1995 - they're very nice reading as well.

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Nice! fillerfiller –  Alex Youcis Mar 30 '13 at 5:19
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Here are links to some notes I have found beneficial:

Andrew Baker - Galois Theory

http://www.maths.gla.ac.uk/~ajb/course-notes.html

Theodore Shifrin - Differential Geometry

http://www.math.uga.edu/~shifrin/

Vaughan Jones - Real Analysis (these are a student's transcription)

https://sites.google.com/site/math104sp2011/lecture-notes

Steven Kleiman - Commutative Algebra

http://web.mit.edu/18.705/www/syl12f.html

Svetlana Katok - p-Adic Analysis

http://www.math.psu.edu/katok_s/pub/p-adic.pdf

And here are several by Paul Garrett who is an active participant here:

http://www.math.umn.edu/~garrett/

Analysis Notes, Shlomo Sternberg (Harvard)

http://www.math.harvard.edu/~shlomo/

Notes by Matthew Morrow on the algebraic number theory material in "Ireland & Rosen" Chapters 12 and 13.

http://www.math.uni-bonn.de/people/morrow/242.pdf

James Binney (Oxford) - Great quantum mechanics videos and a free accompanying downloadable book. Not strictly speaking "math," but plenty of it.

http://www.physics.ox.ac.uk/users/Cruickshank/ http://www-thphys.physics.ox.ac.uk/people/JamesBinney/QBhome.htm

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+1 for a very good addition to my and others' list-the Shifrin notes are a gem online and I wasn't aware of the Kleiman notes! –  Mathemagician1234 Mar 3 '13 at 19:58
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Gathmann's notes on algebraic geometry. I think they are one of the best places from where one can start learning algebraic geometry.

http://www.mathematik.uni-kl.de/~gathmann/alggeom.php

Lecture notes on complex representation theory of finite groups, character theory by A Bartel

http://homepages.warwick.ac.uk/staff/A.Bartel/docs/reptheory.pdf

Lecture notes on abstract algebra, galois theory and introductory algebraic number theory by Frederique Oggier

http://www1.spms.ntu.edu.sg/~frederique/Teaching.html

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For Spanish speaking users, I can link to Carlos Ivorra's website. He has the following material:

  • Logic and set theory
  • Consistency tests
  • Set theory
  • Descriptive set theory
  • Non-standard analysis
  • Algebra
  • Geometry
  • Analysis
  • Functions of complex variable
  • Number Theory
  • Class Field Theory
  • Algebraic Topology
  • Algebraic Geometry
  • Algebraic Curves
  • Homological Algebra and Commutative algebra
  • Representation of finite groups
  • Schemes
  • Arithmetical surfaces
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7  
I wish I spoke Spanish :/ –  Alex Youcis Mar 8 '13 at 3:18
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Zev Chonoles, a graduate student at the University of Chicago and sometime poster here, has several wonderful sets of lecture notes of the first year graduate courses at U of Chicago. They're terrific and strongly recommended.

E. Kowalski of ETH Zurich in Germany has some very good,substantial notes at his webpage on analysis, representation theory and graph theory research. Worth a look.

Alex, you'll be happy to know I've compiled a vast list of lecture notes,with my commentary on each.This list contains just about all the notes at this thread and many more. It will be part of a fully functional website affiliated with my blog, "Tables, Chairs and Beermugs" and will attempt to catalog and comment on all currently available sources for mathematics students under 30 dollars U.S. and the use of links to lecture notes figures very heavily into it, as well as all Dover paperbacks I've read and cheaply printed self-printed books,of which there is a slowly growing number. I'll post a link to the site here when the first version is ready and online.

In a way,I'm kind of upset that you posted this question and the links beating me to it. Oh well, I'm putting mine up anyway.

Consider these to be a preview.

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+1 for Kowalski's notes. But ETH Zürich isn't in Germany, it is in Switzerland! –  Dave Hartman Mar 8 '13 at 8:22
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@Dave Oopsie. : ( –  Mathemagician1234 Mar 10 '13 at 22:30
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Two sets of lecture notes on category theory that have helped me a lot:

Notes taken by Richard Garner on a course by Eugenia Cheng.

Notes by Tom Leinster.

They don't cover exactly the same topics. They are quite different: the first one is more to-the-point, with less details but with some great explanations and insights. It covers a bit more material, if I'm not mistaken. For example, it has proofs for both adjoint functor theorems, and they are very nice, especially the one for the GAFT, in my opinion.

The second one is more verbose, taking the time to fully explain concepts which are seldom really explained in books (e.g. Yoneda's lemma), with great examples.

Both are great and compliment each other really nicely.

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