Best Sets of Lecture Notes and Articles

Question

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$3.$ Often times books are very outdated.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

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

17. 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.).

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

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

Model Theory:

1. The Model Theory of Fields-Marker

Number Theory:

2. Algebraic Number Theory-Conrad

3. Algebraic Number Theory-Weston

4. Class Field Theory-Lemmermeyer

5. Compilation of Notes from Things of Interest to Number Theorists

6. Elliptic Modular Forms-Don Zagier

7. Modular Forms-Martin

8. What is a Reciprocity Law?-Wyman

9. Class Field Theory Summarized-Garbanati

10. Three Lectures About the Arithmetic of Elliptic Curves-Mazur

11. Congruences Between Modular Forms-Calegari

12. Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture-Rubin

13. Simple Proof of Kronecker Weber-Ordulu

14. Tate's Thesis-Binder

15. Introduction to Tate's Thesis-Leahy

16. [A Summary of CM Theory of Elliptic Curves-Getz]

17. An Elementary Introduction to the Langland's Program-Gelbart

18. $p$-adic Analysis Compared to Real Analysis-Katok (Contributed by Andrew; no longer on-line - but here is a snapshot from the Wayback Machine)

19. Representation of $p$-adic Groups-Vinroot

20. Counting Special Points: Logic, Diophantine Geometry, and Transcendence Theory-Scanlon

21. Algebraic Number Theory-Holden

22. The Theory of Witt Vectors-Rabinoff

Complex Geometry:

22. Complex Analytic and Differential Geometry-Demailly

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

24. Uniformization Theorem-Chan

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

26. Complex Manifolds-Koppensteiner

27. Kahler Geometry and Hodge Theory-Biquard and Horing

28. Kahler Geometry-Speyer

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:

2. Foundations of Algebraic Geometry-Vakil

3. Analytic Techniques in Algebraic Geometry-Demailly

4. Algebraic Geometry-Gathmann (Contributed by Mohan)

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

6. Galois Theory for Schemes-Lenstra

7. Rational Points on Varieties-Poonen

8. Teaching Schemes-Mazur

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

Answer 1

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.

Answer 2

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://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

Jerry Shurman - Multivariate Calculus, Complex Ananylis, Algebra, Number Theory

http://people.reed.edu/~jerry/

Alistair Savage - Lie THeory ( based on Stillwell's and Hall' texts)

http://alistairsavage.ca/mat4144/notes/MAT4144-5158-LieGroups.pdf

Answer 3

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

https://www.maths.gla.ac.uk/~abartel/docs/reptheory.pdf

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

https://feog.github.io/

Answer 4

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

Answer 5

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 Switzerland 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.

Answer 6

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.

Answer 7

Undergraduates in Cambridge often refer to the notes of a student that typeset a lot of the lectures when he was there:

https://dec41.user.srcf.net

This includes detailed notes for the courses from first to fourth year.

Answer 8

The Oxford University Maths department seem to have their full set of lecture notes with example sheets on here:

https://www0.maths.ox.ac.uk/courses/material/

Answer 9

I will recommend Dr. Merry's notes. They are fantastic and very user friendly to be honest.

You can find them in here: http://merry.io

Answer 10

I recommend two websites:
a)MIT OCW Mathematics for free lecture notes. Videos, notes, problem sets, etc. are all available.
b)Wolfram mathworld for articles. It has information on almost every topic.