Question

I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable because of writing quality, organization or presentation.

I recently found out that "The Elements of Statistical Learning' by Hastie, Tibshirani and Friedman was available free online: http://www-stat.stanford.edu/~tibs/ElemStatLearn/ . It is a really well written book at a high technical level. Moreover, this is the second edition which means the book has already gone through quite a few levels of editing.

I was quite amazed to see a resource like this available free online.

Now, my question is, are there more resources like this? Are there free mathematics books that have it all: well-written, well-illustrated, properly typeset and so on?

Now, on the one hand, I have been saying 'book' but I am sure that good mathematical writing online is not limited to just books. On the other hand, I definitely don't mean the typical journal article. It's hard to come up with good criteria on this score, but I am talking about writing that is reasonably lengthy, addresses several topics and whose purpose is essentially pedagogical.

If so, I'd love to hear about them. Please suggest just one resource per comment so we can vote them up and provide a link!

Answer 1

John Baez's stuff is a fantastic resource for learning about - well, whatever John Baez is interested in, but fortunately that's a lot of interesting stuff. Scroll down for a link to TWF as well as his expository articles.

Answer 2

The wonderful webpage of Milne has books/lectures notes on a wide variety of topics, including Algebraic geometry, Etale cohomology, Class field theory, ...

Answer 3

Everybody probably knows about this already, but Allen Hatcher's textbook on Algebraic Topology is excellent - clear, well-written, neatly typeset. It takes the student from basic concepts like homotopy equivalence all the way through to things like higher homotopy groups, obstruction theory and representability.

(His partially-written books on K-Theory and Spectral Sequences are also worth a look.)

Answer 4

The second edition of generatingfunctionology by Herbert Wilf is freely available online and is one of my favorite math books ever. It's one of the books that made me fall in love with combinatorics (the other being the Bollobas Graph Theory book).

Answer 5

The wonderful book "A=B", by Marko Petkovsek, Herbert Wilf and Doron Zeilberger, is available freely online, thanks to their publisher AK Peters.

If you've ever wondered how to prove identities for q-multinomials and friends, well, the summary of this book is that computers now know how to do it, and you shouldn't bother anymore.

Answer 6

I hope it's not too rude to double-post, but as far as high-quality books go, Fulton's Algebraic Curves was also recently made available online.

Answer 7

And just because I like the book so much, Flajolet and Sedgewick's Analytic Combinatorics is available online and is a great resource for learning about asymptotic analysis in combinatorics. The first half is also a great introduction to various techniques for writing down generating functions.

Answer 8

I'm sort of surprised nobody has mentioned Terry Tao's blog yet. I think it definitely belongs in this list.

Answer 9

Many know Hatchers Book, but few know the nice Concise Course in Algebraic Topology by J.P.May, which discusses, aside the standard stuff, Groupoids, Higher Homotopy and all that in a very brief and modern fashion. I think this is the book to read (for free) after/between Hatchers book.

There is also a big literature overview included, at the end of the book.

May has written much more (just look at his homepage), and I didn't read all of it. But what I read, I liked.

Answer 10

Check out Allen Hatcher's online books (topological stuff).

Answer 11

Volumes 28 through 56 of the MSRI book series are available here:

http://www.msri.org/communications/books/index.html

Answer 12

Diestel's Graph Theory is probably not as canonical as Hatcher's textbook, but it's a very commonly used textbook for graduate courses in the subject, and it's a similarly broad basic reference.

Answer 13

Abstract and concrete categories: The joy of cats by Jiri Adamek, Horst Herrlich and George Strecker, is a nice book for learning category theory. It went out of print, so the authors made it available online for free.

Answer 14

Robert Ash is a professor who's in the habit of making his textbooks available online as well.

Answer 15

A few more recommendations, apparently less well-known:

Saha's "Principles of Data Analysis" (you seem to have an interest in that field)

Noam Elkies' Lecture Notes (e.g this one on Analytic Number Theory) are like small books.

"Algorithmic Game Theory" by Nisan, Roughgarden, Tardos and Vazirani.

Answer 16

MAA Writing Awards

Answer 17

"Linear Algebra" by Jim Hefferon has been online for a while and it was what I used to teach myself linear algebra. It's very well written with tons of great practice problems and interesting asides. It is a little less advanced than any of the other books listed so far, but it's still a great read. Plus, it's open source (You can download the LaTeX for the book from the website).

Answer 18

I recommend Mel Hochster's notes. The notes for Math 614 and 615 form an introduction to commutative algebra, and 711 is on a different topic (tight closure, Henselization, etc.) every year. I think they're very easy to read.

Answer 19

IF you want to see free academic video courses from leading universities, just go to

http://www.academicearth.org/

Answer 20

Information Theory, Inference, and Learning Algorithms by David MacKay of the Cavendish Laboratory.

Answer 21

The two-volume relatively introductory work on operator algebras, Operator Algebras and Quantum Statistical Mechanics by O. Bratteli and D. Robinson is available at Bratteli's website.

Answer 22

http://www.cs.utk.edu/~dongarra/etemplates/book.html

Answer 23

Len Evens has a couple of online textbooks: a text on abstract algebra, and a linear algebra text.

Answer 24

Martin J Osborne and Ariel Rubinstein's book "A course in Game theory" is available here.

Answer 25

Stephen Boyd has some good books on his Stanford home page: http://www.stanford.edu/~boyd/books.html ... especially the one on convex optimization is very good.

Answer 26

Not really pointing to a book, but I'd like to let you know I'm soon (within a month or so) launching a site dedicated to this. It is now almost finished. It is going to be a place where people can add mathematical resources, vote on them, add reviews, see other people's favorites and so on. Books will be categorized by language, level, topics, status (draft, lecture notes, books) and so on. I hope I will be able to "advertise" it trough mathoverflow: as with many "social" sites, the more people join, the more interesting it will become.

Answer 27

'The Elements of Statistical Learning' by Hastie, Tibshirani and Friedman http://www-stat.stanford.edu/~tibs/ElemStatLearn/

Answer 28

The Caltechbook service at Caltech offers a number of math books for free here, including some very good (IMHO) books by Jerry Marsden et al.

Answer 29

And another great source of the lecture notes and stuff is the MIT OpenCourseWare, in particular the math section.

Answer 30

As with so many things, "There's a reddit for that": http://www.reddit.com/r/mathbooks. It's a mixed bag—much like searching for math books in a non-specialist bookstore, one gets the elementary mixed up with the sophisticated—but there are some gems there.