User:IkamusumeFan/sandbox/High Quality Online Resources for Advanced Math

This page summarizes a list of great online resources for advanced mathematics. I hope this can be your start to find good stuff to learn within advanced mathematics!

Further updates will follow in the future.

Existing Collections
George Cain from Gatech has given a great summary of online math textbooks. It has not been updated from 2014, though. Robert Ash's textbooks, including algebras, complex variables, probability, etc. J.S. Milne's course notes.
 * George Cain's collection of great online math textbooks.
 * Robert Ash's textbooks.
 * Milne's lecture notes.

Algebra
John Scherk's intro to algebra from computation aspect. Pretty like Victor Shoup's work. Leonard Evens's notes used by Northwestern University. Some notes from Jonathan Brundan.
 * Computational introduction to algebra.
 * Algebra at Northwestern University.
 * Algebra in U of Oregon.

Algebraic Geometry
Kiran Kedlaya wrote some lecture notes about AG in MIT. They can help you while reading Hartshorne's book! Igor Dolgachev has written many notes related to AG, e.g. modular forms, scheme, cats, etc. Kimball Martin wrote some notes on modular forms, and he would like to maintain them.
 * Kiran Kedlaya's lecture notes.
 * Igor Dolgachev's lecture notes.
 * Kimball Martin's lecture notes on modular forms.

Algebraic Topology
James Davis's notes. May refer while reading Allen Hatcher's. J. P. May's concise notes.
 * Lecture Notes in Algebraic Topology
 * Concise AT.

Category Theory
David Spivak introduced cats to non-mathematicians! So his notes should be quite readable.
 * David Spivak's Categories for Scientists.

Complexity Theory
Jonathan Katz has given great lecture notes. Sanjeev Arora and Boaz Barak have kindly given the draft of their influential textbook. Daniel Spielman has taught an advanced course on complexity theory in MIT. Great lecture notes (including crypto) are given by Luca Trevisan. Oded Goldreich's notes on computation complexity are at the end of the below webpage. There are also many great crypto books. Ryan O'Donnell has some high-quality lectures on computation, e.g., PCP, LP, Boolean, Quantum, etc. Madhu Sudan also has some.
 * Jonathan Katz's lecture notes on complexity theory.
 * Draft for Computational Complexity: A Modern Approach.
 * Daniel Spielman's Advanced Complexity Course.
 * Luca Trevisan's lecture notes on complexity.
 * Goldreich's books and lecture notes.
 * See Prior teaching (lecture notes and videos).
 * Complexity, Coding Theory, etc.

Cryptography
Alessandro Chiesa gives the most comprehensive collection of online resources I have ever seen! Yevgeniy Dodis has given two courses on advanced crypto topics. The below one focused on personal data extractions. The next one focused on zero knowledge protocols. Jonathan Katz has proofreaded his advanced crypto lecture notes. Again, they focused on zero knowledge. Dan Boneh and Victor Shoup's textbook. It is still being updated. Some later sections might be missing.
 * Alessandro Chiesa's collection of crypto notes.
 * Randomness in Cryptography.
 * Advanced Cryptography.
 * Advanced Topics in Cryptography.
 * A Graduate Level Crypto Intro.

Descriptive Set Theory
UCLA's version is quite comprehensive. UIC's is more concise. Anush Tserunyan's version is continuously developed. Clearly it is not done now.
 * UCLA DST Lecture Notes
 * Concise notes from UIC
 * UIUC Notes on DST. Not done yet.

General Topology
Sidney A. Morris writes an introductory book to topologies. So far, this book is actively updated.
 * Topology without Tears.

Graph Theory
Daniel Spielman has given lectures on spectral graph theory for many years (maybe since 2004). You can also check the old notes on his website.
 * Spielman's lecture notes on Spectral Graph Theory.

Lattice-based Cryptography
Oded Regev's notes: Shai Halevi's notes (with Tal Malkin):
 * Regev 2004 in NYU.
 * Lattice crypto in Columbia University.
 * Hosted in MIT.

Model Theory
Anush Tserunyan's notes on logics and incompleteness theorems.
 * Still under development.

Privacy
Aaron Roth's lectures contain the full textbook authored with Cynthia Dwork.
 * Privacy Lectures @UPenn.

Probability Theory
We appreciate Scott Sheffield's summary!
 * Scott Sheffield's collection.

Quantum Computing
Quantum computing courses from Caltech. A course by Ryan O'Donnell and John Wright from CMU. Also Robert Griffiths from physics has some. Umesh Varizani's notes in Berkeley. Dave Bacon taught a course in UW.
 * Quantum Computation in Caltech.
 * Quantum Computation and Information 2015.
 * Robert Griffiths's notes.
 * Quantum Computation in Berkeley.
 * Quantum Computing in UW.

Random Processes
Aldous and Fill have a book on Markov Chains and Random Walk. This book is still under development, quite slowly though.
 * Aldous and Fill's monograph on Reversible Markov Chains and Random Walks on Graphs

Randomization
Salid Vadhan's monograph on pseudorandomness. Oded Goldreich's randomized computation. Also Yevgeniy Dodis discussed randomness in crypto (see his notes in Cryptography section).
 * Salid Vadhan's Pseudorandomness.
 * Randomized Methods in Computation.