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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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There is a whole database of freely available books at e-booksdirectory.com/mathematics.php; I'm making this a comment since it's not one specific volume. –  Akhil Mathew Oct 21 '09 at 21:26
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There are several links, both to specific free books and to databases, in a related question: mathoverflow.net/questions/761?sort=votes#sort-top –  Qiaochu Yuan Oct 21 '09 at 21:30
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A list which I don't think is mentioned in the other question is here: people.math.gatech.edu/~cain/textbooks/onlinebooks.html –  Qiaochu Yuan Oct 21 '09 at 21:37
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Please, add them to mathonline.andreaferretti.it too. :-) –  Andrea Ferretti Nov 19 '10 at 14:47
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54 Answers

up vote 27 down vote accepted

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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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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And since JS Milne has become a MathOverflow user: A big Thank you! - just in case you are reading this... –  Peter Arndt Oct 21 '09 at 22:06
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I have a lot of respect for mathematicians such as J.S. Milne and Allen Hatcher who disseminate their work widely and freely. –  Beren Sanders Nov 19 '10 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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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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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." –  John Sidles May 25 '11 at 19:20
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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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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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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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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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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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http://mathunion.org/ICM/ has almost all volumes of ICM talks online

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Starting from 1893! –  Joseph O'Rourke Dec 21 '10 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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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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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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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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@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. ;) –  Quadrescence Nov 21 '10 at 20:52
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@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. –  Andy B Nov 21 '10 at 21:13
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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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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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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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IF you want to see free academic video courses from leading universities, just go to

http://www.academicearth.org/

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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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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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I'm glad someone mentioned Keith Conrad's notes, as they are excellent.

I would also like to point people towards Tom Weston's webpage. He has expository papers at http://www.math.umass.edu/~weston/ep.html on several topics, including cobordism theory and spectral sequences.

He also has some course notes at http://www.math.umass.edu/~weston/cn.html, including truly excellent book-length notes on introductory algebraic number theory, as well as several dozen illuminating pages on local fields and ideles.

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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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Check out Jean-Pierre Demailly's books on analytic algebraic geometry http://www-fourier.ujf-grenoble.fr/~demailly/books.html.

Here you go the AMS book online webpage http://www.ams.org/online_bks/online_subject.html .

I should also mention the AMS online book webpage collection http://www.ams.org/online_bks/online-books-web.html.

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