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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!

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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.

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  • $\begingroup$ A million times seconded. John Baez has an incredible knack for explaining very abstract stuff in a very intuitive way; "Read John Baez!" has become something of a mantra of mine lately. $\endgroup$
    – Harrison Brown
    Commented Oct 21, 2009 at 21:40
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The wonderful webpage of Milne has books/lectures notes on a wide variety of topics, including Algebraic geometry, Etale cohomology, Class field theory, ...

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    $\begingroup$ And since JS Milne has become a MathOverflow user: A big Thank you! - just in case you are reading this... $\endgroup$
    – Peter Arndt
    Commented Oct 21, 2009 at 22:06
  • $\begingroup$ Seconded to JS Milne! $\endgroup$
    – Akhil Mathew
    Commented Oct 22, 2009 at 1:46
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    $\begingroup$ I have a lot of respect for mathematicians such as J.S. Milne and Allen Hatcher who disseminate their work widely and freely. $\endgroup$
    – Beren Sanders
    Commented Nov 19, 2010 at 8:08
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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.)

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

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  • $\begingroup$ It certainly does,Darsh. : ) $\endgroup$
    – The Mathemagician
    Commented May 25, 2011 at 23:36
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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).

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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.

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    $\begingroup$ The preface that Donald Knuth wrote for A=B is terrific too! "Science is what we understand well enough to explain to a computer. Art is everything else we do. [\ldots] Science advances whenever an Art becomes a Science. And the state of the Art advances too, because people always leap into new territory once they have understood more about the old." $\endgroup$
    – John Sidles
    Commented May 25, 2011 at 19:20
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I was hoping that someone had posted Keith Conrad's expository stuff. Twice this week I've searched for an example in algebraic number theory (it is somewhat surprising how few of these there are in the books I own) and found the perfect answer on that page. The papers are remarkable for their high number of carefully chosen examples, just enough of which are worked out for the reader.

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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.

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  • $\begingroup$ A true classic with a sequel on the way,as reported here a few months ago. $\endgroup$
    – The Mathemagician
    Commented May 24, 2010 at 5:06
  • $\begingroup$ But that's even too difficult for grad course in a semester... $\endgroup$
    – Henry.L
    Commented May 10, 2013 at 10:34
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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.

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  • $\begingroup$ Fantastic book! $\endgroup$
    – GMRA
    Commented Oct 21, 2009 at 22:10
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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.

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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.

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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.

EDIT: The site is now online. It's still young, but I hope it will improve with time; I certainly have to add some features, but I decided it was time to launch and see if people actually find it useful. You can find it here

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  • $\begingroup$ The link seems to be dead. $\endgroup$
    – I. J. Kennedy
    Commented Jul 24, 2013 at 22:44
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http://mathunion.org/ICM/ has almost all volumes of ICM talks online

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    $\begingroup$ Starting from 1893! $\endgroup$
    – Joseph O'Rourke
    Commented Dec 21, 2010 at 17:32
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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.

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I can't believe nobody mentioned : NUMDAM and Göttinger Digitalisierungszentrum, where you'll find digitized versions of mathematical texts... monographies and articles which made mathematical history, but sometimes still count as important references!

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

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  • $\begingroup$ There's already a different answer about Hatcher's writings. $\endgroup$
    – David White
    Commented Feb 14 at 0:35
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Volumes 28 through 56 of the MSRI book series are available here:

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

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Paul Garrett is quite the author:

http://www.math.umn.edu/~garrett/

He has a book on buildings and many vignettes about automorphic forms, L-functions, representation theory, .... He wrote a graduate algebra book while he taught the course, and promptly got it published.

http://www.math.umn.edu/~garrett/m/algebra/

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  • $\begingroup$ It's not a bad book,Garrett's-but it's really hard to consider an algebra text in 2010 that doesn't do category theory or homological algebra AT ALL a GRADUATE algebra text. $\endgroup$
    – The Mathemagician
    Commented Nov 19, 2010 at 17:27
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    $\begingroup$ @Andrew L: "General Algebra" at the University of Minnesota covers what undergrad algebra would cover, but at a greater depth. I'd say it's sort of like the difference between "calculus" and "advanced calculus". Also, Garrett makes/grades algebra prelim exams IIRC, so the book is very good preparation. Lastly, I must say he's a very nice guy. He has a very pleasing ideology on mathematics and education (very "I want you to learn" attitude, not "I want you to get a good grade". in fact, in his $L$-functions and automorphic forms class, you get an "A", but you're still required to do work. ;) $\endgroup$
    – Quadrescence
    Commented Nov 21, 2010 at 20:52
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    $\begingroup$ @AndrewL: Garrett's book, like Garrett, is unconventional. As mentioned above, the point is not to do as much algebra as is feasible in a year, the point is to do a good amount of algebra from the "right" (in the Garrett sense, whatever that means) perspective. Most mathematicians do not use category theory or homological algebra at all, and I find a first year graduate text on algebra being devoid of these topics as no great sin. Besides, only a foolish graduate student uses one algebra book. $\endgroup$
    – Andy B
    Commented Nov 21, 2010 at 21:13
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Robert Ash is a professor who's in the habit of making his textbooks available online as well.

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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.

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MAA Writing Awards

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In nlab we keep a list of main links of archives and free book collections in our main areas of interests (we were intentionally selective there):

For top level directory for math resources see http://ncatlab.org/nlab/show/math+resources, from where you can go to archives, individual author collections, blogs and institutions.

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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.

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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.

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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.

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

http://www.academicearth.org/

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

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

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A great hidden gem is Shlomo Sternberg's page of online books:

http://www.math.harvard.edu/~shlomo/

Also, Curt McMullen has some notes at the bottom of this page

http://www.math.harvard.edu/~ctm/papers/index.html

which are good, but less formal. He also has other notes on his website not listed there; just look at his list of past courses and follow the links.

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  • $\begingroup$ Sternberg's notes are terrific,but pretty well knowm.Curtis McMullen is quietly one of the best teachers among prominent mathematical researchers and his lecture notes are a great addition to the available free online resources.They are fairly sophisticated,though-as would be expected from Harvard coursework-so not for beginners. $\endgroup$
    – The Mathemagician
    Commented Nov 19, 2010 at 17:34
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The Stacks Project http://math.columbia.edu/algebraic_geometry/stacks-git/

If I ever wonder about something, I can pretty much count on it to be in there. Remember to use a PDF viewer with hyperlinks and back/next buttons.

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