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.

• George Cain's collection of great online math textbooks.

Robert Ash's textbooks, including algebras, complex variables, probability, etc.

• Robert Ash's textbooks.

J.S. Milne's course notes.

• Milne's lecture notes.

Algebra

John Scherk's intro to algebra from computation aspect. Pretty like Victor Shoup's work.

• Computational introduction to algebra.

Leonard Evens's notes used by Northwestern University.

• Algebra at Northwestern University.

Some notes from Jonathan Brundan.

• 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!

• Kiran Kedlaya's lecture notes.

Igor Dolgachev has written many notes related to AG, e.g. modular forms, scheme, cats, etc.

• Igor Dolgachev's lecture notes.

Kimball Martin wrote some notes on modular forms, and he would like to maintain them.

• Kimball Martin's lecture notes on modular forms.

Algebraic Topology

James Davis's notes. May refer while reading Allen Hatcher's.

• Lecture Notes in Algebraic Topology

J. P. May's concise notes.

• 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.

• Jonathan Katz's lecture notes on complexity theory.

Sanjeev Arora and Boaz Barak have kindly given the draft of their influential textbook.

• Draft for Computational Complexity: A Modern Approach.

Daniel Spielman has taught an advanced course on complexity theory in MIT.

• Daniel Spielman's Advanced Complexity Course.

Great lecture notes (including crypto) are given by Luca Trevisan.

• Luca Trevisan's lecture notes on complexity.

Oded Goldreich's notes on computation complexity are at the end of the below webpage. There are also many great crypto books.

• Goldreich's books and lecture notes.

Ryan O'Donnell has some high-quality lectures on computation, e.g., PCP, LP, Boolean, Quantum, etc.

• See Prior teaching (lecture notes and videos).

Madhu Sudan also has some.

• Complexity, Coding Theory, etc.

Cryptography

Alessandro Chiesa gives the most comprehensive collection of online resources I have ever seen!

• Alessandro Chiesa's collection of crypto notes.

Yevgeniy Dodis has given two courses on advanced crypto topics. The below one focused on personal data extractions.

• Randomness in Cryptography.

The next one focused on zero knowledge protocols.

• Advanced Cryptography.

Jonathan Katz has proofreaded his advanced crypto lecture notes. Again, they focused on zero knowledge.

• Advanced Topics in Cryptography.

Dan Boneh and Victor Shoup's textbook. It is still being updated. Some later sections might be missing.

• A Graduate Level Crypto Intro.

Descriptive Set Theory

UCLA's version is quite comprehensive.

• UCLA DST Lecture Notes

UIC's is more concise.

• Concise notes from UIC

Anush Tserunyan's version is continuously developed. Clearly it is not done now.

• 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:

• Regev 2004 in NYU.

Shai Halevi's notes (with Tal Malkin):

• 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.

• Quantum Computation in Caltech.

A course by Ryan O'Donnell and John Wright from CMU.

• Quantum Computation and Information 2015.

Also Robert Griffiths from physics has some.

• Robert Griffiths's notes.

Umesh Varizani's notes in Berkeley.

• Quantum Computation in Berkeley.

Dave Bacon taught a course in UW.

• 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.

• Salid Vadhan's Pseudorandomness.

Oded Goldreich's randomized computation.

• Randomized Methods in Computation.

Also Yevgeniy Dodis discussed randomness in crypto (see his notes in Cryptography section).