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

Within the framework of the project retro.seals.ch, scientific journals are retrodigitized and made available via internet. The project contains the following mathematical journals:

• Commentarii Mathematici Helvetici
• Elemente der Mathematik
• Elemente der Mathematik (Beihefte zur Zeitschrift)
• L'Enseignement Mathématique

Answer 3

The Comptes Rendus de l'Académie des Sciences are available online.

Answer 4

The website of the Leibniz award offers a free online collection: link, and there is the preprint server of the IHES.

Answer 5

Gerald Teschl books

1. Textbook Ordinary Differential Equations and Dynamical Systems
2. Textbook Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators

can be found at http://www.mat.univie.ac.at/~gerald/

Answer 6

Richard P. Stanley's Enumerative Combinatorics, volume 1, second edition is available at http://www-math.mit.edu/~rstan/ec/ec1/ .

Answer 7

Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths is available through MSRI (and is sadly out of print at the moment).

http://library.msri.org/books/masterlist.html

Answer 8

Book by David Levin, Yuval Peres and Elizabeth Wilmor on Markov chain theory and mixing times. http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf. It quickly takes someone with basic knowledge in probability and linear algebra into the heart of current research.

Answer 9

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!

Answer 10

http://mathunion.org/ICM/ has almost all volumes of ICM talks online

Answer 11

You can find here large pool of ebooks in every brach of Math.

Answer 12

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.

Answer 13

http://www.oocities.com/alex_stef/mylist.html

Answer 14

Basic Concepts of Enriched Category Theory by G.M. Kelly was originally published by Cambridge University Press in 1982 but is now available online: http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html

My understanding is that it is the canonical reference for enriched category theory (and was written by the pioneer of the field).

Answer 15

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.

Answer 16

Jim Pitman's Combinatorial Stochastic Processes.

Later edit: http://works.bepress.com/jim_pitman/1

Answer 17

Jerome Keisler's Elementary Calculus. This book uses infinitesimals explicitly, and also in a logically rigorous way, without getting too advanced for first-year undergraduates.

Later edit: http://math.wisc.edu/~keisler/calc.html

Answer 18

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.

Answer 19

Roland Speicher has some nice introductory material for Free Probability (mini course, survey articles etc.) All available at http://www.mast.queensu.ca/~speicher/survey.html

Answer 20

Sergei Winitzki has an interesting-looking book on the coordinate-free approach to linear algebra online.

Answer 21

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/

Answer 22

A draft of Albert Marden's Outer circles: an introduction to hyperbolic 3-manifolds is online, on his website:

http://www.math.umn.edu/~am/book/outercircles.pdf

Edward Nelson's Radically elementary probability theory is also online, on his website:

http://www.math.princeton.edu/~nelson/books/rept.pdf

Answer 23

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.

Answer 24

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.

Answer 25

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

Answer 26

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.

Answer 27

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.

Answer 28

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

Answer 29

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 30

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.