Best Sets of Lecture Notes and Articles

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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$3.$ Often times books are very outdated.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

  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:

  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; no longer on-line)
  18. Representation of $p$-adic Groups-Vinroot
  19. Counting Special Points: Logic, Diophantine Geometry, and Transcendence Theory-Scanlon

  20. Algebraic Number Theory-Holden

  21. The Theory of Witt Vectors-Rabinoff

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

  6. Kahler Geometry and Hodge Theory-Biquard and Horing

  7. 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:

  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

  7. 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: http://www.pdfmerge.com/

Answer

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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