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$3.$ Often times books are very outdated.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:

Andrew Baker (Contributed by Andrew)

Garrett (Contributed by Andrew)

Frederique (Contributed by Mohan)

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

ALGANT Masters Theses (an absolutely stupendous collection of masters theses in various aspects of algebraic geometry/algebraic number theory).

The Stacks Project (an open source ‘textbook’ with the goal in mind to have a completely self-contained exposition of the theory of stacks. Because such a huge amount of background is required, it contains detailed articles about commutative algebra, homological algebra, set theory, topology, category theory, sheaf theory, algebraic geometry, etc.).

Harvard undergraduate theses (an excellent collection of the mathematics undergraduate theses completed in the last few years at Harvard).

Bas Edixhoven (this is a list of notes from talks that Edixhoven has given over the years).

Model Theory:

Number Theory:

- Algebraic Number Theory-Conrad
- Algebraic Number Theory-Weston
- Class Field Theory-Lemmermeyer
- Compilation of Notes from Things of Interest to Number Theorists
- Elliptic Modular Forms-Don Zagier
- Modular Forms-Martin
- What is a Reciprocity Law?-Wyman
- Class Field Theory Summarized-Garbanati
- Three Lectures About the Arithmetic of Elliptic Curves-Mazur
- Congruences Between Modular Forms-Calegari
- Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture-Rubin
- Simple Proof of Kronecker Weber-Ordulu
- Tate’s Thesis-Binder
- Introduction to Tate’s Thesis-Leahy
- [A Summary of CM Theory of Elliptic Curves-Getz]
- An Elementary Introduction to the Langland’s Program-Gelbart
~~$p$-adic Analysis Compared to Real Analysis-Katok~~(Contributed by Andrew; no longer on-line)- Representation of $p$-adic Groups-Vinroot
- Counting Special Points: Logic, Diophantine Geometry, and Transcendence Theory-Scanlon

Complex Geometry:

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

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

Differential Topology/Geometry:

Lie Groups-Ban (comes with accompanying lecture videos)

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

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

Algebra:

Category Theory-Leinster (Contributed by Bruno Stonek)

Category Theory-Chen (Contributed by Bruno Stonek)

Commutative Algebra-Altman and Klein (Contributed by Andrew)

Finite Group Representation Theory-Bartel (Contributed by Mohan)

Topology

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

Algebraic Geometry:

Algebraic Geometry-Gathmann (Contributed by Mohan)

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/

**Answer**

In no particular order:

- Algebraic number theory notes by Sharifi: http://math.arizona.edu/~sharifi/algnum.pdf
- Dalawat’s first course in local arithmetic: http://arxiv.org/abs/0903.2615
- Intro to top grps: http://www.mat.ucm.es/imi/documents/20062007_Dikran.pdf
- Representation theory resources: http://www.math.columbia.edu/~khovanov/resources/
- Classical invariant theory: http://jones.math.unibas.ch/~kraft/Papers/KP-Primer.pdf
- CRing project: http://people.fas.harvard.edu/~amathew/CRing.pdf – The notes are huge & has many authors – including MSE’s Zev, Akhil (no longer active) & Darij. Check the ToC.
- Partitions bijections, a survey: http://www.math.ucla.edu/~pak/papers/psurvey.pdf
- Hidden subgroup problem (review, open stuff): http://arxiv.org/abs/quant-ph/0411037
- Spirit of moonshine: http://www.math.harvard.edu/theses/senior/booher/booher.pdf
- Vertex operator algebras and modular forms: http://arxiv.org/abs/0909.4460
- Categorified algebra & quantum mechanics: http://arxiv.org/abs/math/0601458
- Exponential sums over finite fields: http://www.math.ethz.ch/~kowalski/exp-sums.pdf
- Gauss sums: http://math.mit.edu/~brubaker/houghexpsums05.pdf
- Adeles over $\Bbb Q$: https://www.maths.nottingham.ac.uk/personal/ibf/text/gl1.pdf, followed by automo reps over GL(1,A) https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf
- Invariant thry: http://www.win.tue.nl/~jdraisma/teaching/invtheory0910/lecturenotes11.pdf
- Species: http://www.newton.ac.uk/programmes/CSM/Abstract3/Species_intro.pdf
- FLT: http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf
- Categorical concepts: http://www.math.harvard.edu/~eriehl/266x/survey.pdf
- Groups, Rings, Fields (Lenstra): http://websites.math.leidenuniv.nl/algebra/topics.pdf, which is part of algebra notes: http://websites.math.leidenuniv.nl/algebra/

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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