Open/publicly available textbooks worth their salt

I've been reading a bit about "open textbooks", i.e. textbooks made available for easy, online access. These can be nice for those without access to a great library, or who might not be willing to drop the big bucks on a math textbook. In a similar vein, many classic textbooks are available online (legitimately) for free; for example, one can find Oxtoby's Measure and Category in its entirety on Rice's website (I presume that Rice checked that this was kosher before they put it up).

My issue with the open textbooks is that they tend to be very introductory. You have your typical high school maths, e.g. pre-calculus, calculus, and then some introductory texts to abstract algebra, analysis, etc. But if you're interested in anything more, then you're stuck looking for an $80 book, which is typically where they tend to be (from my understanding) more expensive and harder to find. That is, I can get a typical undergrad analysis book pretty easily on these open textbook sites, but if I want anything more, say, an intro to measure-theoretic analysis, then I'm stuck looking through the old routes, i.e. library or Amazon.

My question to you: What are some of your favorite textbooks that are legitimately available for free on the Interwebz? More particularly, I'm interested in books on more specialized subjects (i.e. something beyond an intro to abstract algebra/statistics), though if you have some stellar reference for single-variable calculus or something to that effect, then I'd be glad to hear it as well.

Solution 1:

I figure (I hope!) this will be different in 10 years or so, but right now you'll have to combine many sources if you want to get a mathematical education purely from free/open sources; in particular, you'll probably want to include lecture notes in your reading list. I have seen few one-stop shops like Dummit&Foote or Lang or Rudin in the public domain.

Here is a selection of notes and texts I know about. I am not 100% sure if all of the below sources are slated to stay on the internet indefinitely (as opposed to eventually getting removed when being published as a book -- something which sadly still happens on occasion), but many of them are -- some are actually licensed CC or GPL -- and they are all definitely meant to be downloaded and read. From what I can tell, most of them, if not all, have been written by authors competent in their subjects (to say the least). Are they all reader-friendly and pedagogically valuable? I don't know, but I don't see a reason for them to be less readable than the average printed textbook. It's not like the publishers have figured out the secret of good writing; Lang's textbooks, for example, are famously hard to read due to inaccuracies and sloppy writing, and generally, printed books suffer from restrictions on length, marketing-based editing (van der Waerden's "Modern Algebra" lost one of its most interesting chapters due to the respective theory getting out of fashion), and the difficulty of correcting errors when they are found.

S. Gill Williamson's materials include, in particular, the very well-written (although horribly typed!) Comprehensive Introduction to Linear Algebra by him and Joel G. Broida; combined with the extra materials by Marvin Marcus posted on the same website, these probably cover enough linear algebra for four semesters worth of study. Hefferon provides an introduction to linear algebra that is probably more suitable for the first year, but I suspect that it is an example of the "very introductory" phenomenon the OP is complaining about.

Other materials on Williamson's webpage includes a discrete maths course and a book on combinatorics, both joint with Edward A. Bender (a familiar name if you work in Young tableau theory).

Category theory is particularly well-represented on the internet, maybe because it attracts the same kind of people who work in FOSS. See Awodey's lecture notes, the TAC reprints including, e.g., the Barr-Wells textbook, the Joy of Cats and many other things. Actually, see this list; I am not going to copypaste it in full!

On representation theory, Peter Webb's RepBook and Etingof's maybe-not-quite-introduction. Also lecture notes by Teleman. William Crawley-Boevey has some lecture notes on topics in representations.

Peter J. Cameron's notes include a lot of introductory material, but not only.

For elementary abstract algebra, there is Andrew Baker and James S. Milne.

For commutative algebra: Robert B. Ash, Pete L. Clark, Altman/Kleiman. For an advanced topic, Irena Swanson and Craig Huneke's book on Integral closure.

Jean Gallier's notes, including computer science, algebra and differential geometry.

George M. Bergman on universal algebra.

Keith Conrad on various things, sadly a collection of notes rather than a self-contained text -- but perfect to fill gaps in one's knowledge arising from reading too shallow books. Similarly, Jerry Shurman's materials which seem slightly more systematic, and Patrick Morandi's writing which include "An Introduction to Abstract Algebra via Applications".

The writings of Grothendieck, including the EGA texts, which are still regarded as the complete text on algebraic geometry. Contrast this with Ravi Vakil's 793-page notes, which seem to do a much better job at providing motivation.

Notes by Andreas Gathmann and Thomas Markwig on introductory algebra and algebraic geometry.

Charles Weibel's K-book for K-theory.

Hanspeter Kraft on transformation groups.

Jacob Lurie puts his books online. Please don't misunderstand "higher algebra" as some sequel to basic abstract algebra, though; this is cutting-edge category theory.

Lecture notes by Bodo Pareigis, some of it in English.

Mark Reeder's class pages include notes. Pierre Schapira. Christoph Schweigert. Romyar Sharifi. Michael Stoll. William Stein.

Richard Stanley's Enumerative Combinatorics, volume 1.

Needless to say, the list is incomplete. I have basically browsed through my folder, probably missing half of the good stuff.

Solution 2:

In general, one can find excellent free textbooks or expository articles from Google if one knows what to search. Typically, adding a few keywords from the topic and the word 'pdf', will get you endless resources put on the internet by professors. Or one could type the topic, such as 'Introduction to Algebraic Geometry', and the word 'pdf' and see what comes up. Typically, you will still have to weed through a few pages before you find something you like. However, I would challenge that it is generally difficult to find a topic that does not have a well written pdf placed on the internet by a professor, no matter the level of the topic.

A personal favorite (especially since the book is so funnily written) is The Rising Sea, a textbook on Algebraic Geometry well worth the read - as much as Hartshorne I'm finding! Of course, you have Pete Clark's Commutative Algebra (you can see more of his material here) as another excellent example. If shorter articles on a variety of topics is what you're looking for, try Keith Conrad's page which has endless examples of short well written examples on a large variety of topics.

Furthermore, you can try various Mathematics department's pages. Often, graduate students will put up their talks or class notes (either TeXed or hand written) up on their personal webpage. A personal favorite example of this is Daniel Miller's notes, which he also makes available the TeX file for these - certainly taught me a few things about TeX I did not know!

Even more, if there are not full textbooks or notes on a topic, you might find a well written short article on some mathematicians blog. The most famous example of this would be Terence Tao's blog, which he regularly uses and is well read.

Some professors (as you pointed out) even put all their books online for free! A noteworthy example of this would be Hatcher's Algebraic Topology. That and his other books can be found here.

Again for emphasis, not all of these have to be 'high' level or research level Mathematics. For example, if I wanted to know more about the method of partial fractions when integrating, I'll type 'Partial Fractions Integration pdf' into Google (maybe even add the word examples to the search). This is the second link I found (the first result was great but I appreciated the Simpsons use in the second link):

Integration by Partial Fractions

The trick is learning how to Google well. One you find a page you like, try favoriting it. You never know what mathematical gems might be posted on that page in the future! I hope this was of some help.

EDIT: If you plan on going into Mathematics (which I can only hope is the case), you can even find lots of practice Qualifying exams beyond what your University will offer. An excellent compilation of this has already been done on MathStacks.

EDIT 2: Since I can't but help highlighting a few more of the wonderful gem's you can find online, here are a few more:

Lectures: The Geometry and Algebra of Curves on Surfaces

Blog: Galois Representations

Putnam Archive

Blog: Hard Arithmetic

Robert Ash's Page (with many texts)

OCW Math Page (this has many video lectures, notes, exercises, etc)

Text: Algebraic Number Theory

Harvard PhD Thesis List

And of course, don't forget that YouTube is also a great place to go! There are many excellent lectures and conferences you can watch on there!

HIM Lectures

Institut des Hautes Études Scientifiques (No worries, not in French)

International Centre for Theoretical Sciences

Math Doctor Bob

Of course, you can also find lectures elsewhere. A personal favorite series of lectures of mine to watch would be Peter Scholze's lectures on Perfectoid Spaces.