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[edit]
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.
Algebra[edit]
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.
Algebraic Geometry[edit]
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.
Algebraic Topology[edit]
James Davis's notes. May refer while reading Allen Hatcher's.
J. P. May's concise notes.
Category Theory[edit]
David Spivak introduced cats to non-mathematicians! So his notes should be quite readable.
Complexity Theory[edit]
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.
Cryptography[edit]
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.
Descriptive Set Theory[edit]
UCLA's version is quite comprehensive.
UIC's is more concise.
Anush Tserunyan's version is continuously developed. Clearly it is not done now.
General Topology[edit]
Sidney A. Morris writes an introductory book to topologies. So far, this book is actively updated.
Graph Theory[edit]
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.
Lattice-based Cryptography[edit]
Oded Regev's notes:
Shai Halevi's notes (with Tal Malkin):
Model Theory[edit]
Anush Tserunyan's notes on logics and incompleteness theorems.
Privacy[edit]
Aaron Roth's lectures contain the full textbook authored with Cynthia Dwork.
Probability Theory[edit]
We appreciate Scott Sheffield's summary!
Quantum Computing[edit]
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.
Random Processes[edit]
Aldous and Fill have a book on Markov Chains and Random Walk. This book is still under development, quite slowly though.
Randomization[edit]
Salid Vadhan's monograph on pseudorandomness.
Oded Goldreich's randomized computation.
Also Yevgeniy Dodis discussed randomness in crypto (see his notes in Cryptography section).