Mathematics is a fairly traditional discipline. In contrast to other disciplines, where progress often involves the replacement of one theory by another, mathematics continuously builds on previous knowledge, and the most important tools of a mathematician are still pen and paper, chalk and blackboard.
Despite this natural traditional disposition, the way mathematics is done, learned, published or perceived is, of course, more and more influenced by the internet. The web provides numerous new possibilities, tools and resources – some of them extremely useful, informative or at least entertaining and others (possibly the same) seriously time consuming. This post lists some of them which I use or find interesting. I am curious to find out which ones you use. Are there other links or resources related to mathematics you would recommend? Then let me know …
If you know one mathematics blog, it’s probably Tao’s What’s new. It’s the best example of a really good blog and it’s hard to believe that there is only one person behind it (it seems much more plausible that Tao found a way of teaming up with umpteen of his counterparts from the multiverse). About once a year the content of his blog is published in the form of a book – just in case someone prefers the old look-and-feel.
Gowers’s weblog is another blog you should consider adding to your Reader, if you haven’t already done so. Gowers, like Tao a Fields Medallist, also uses his blog to share his ideas and views on political issues. His recent public announcement to refuse to cooperate with the publisher Elsevier shows what impact a blog post can have: soon after the announcement a remarkable number of (currently >9k) academics also decided to make their protest public and to boycott Elsevier.
Among the science blogs which I read with pleasure are also Gödel’s lost letter and P=NP (the title being a good outline of its content), Shtetl-Optimized (not sure, whether the same about the title holds true here) and backreaction. If you want to browse through the entire mathematical ‘blogosphere’, you should have a look at Mathblogging.org.
Apart from Wikipedia, I occasionally look into Scholarpedia or the Encyclopedia of Mathematics published by Springer in cooperation with the European Mathematical Society. Admittedly, however, I often end up with books in my hands musing about how one could implement an onion-structured encyclopedia which provides information at exactly the right level and depth …
A resource which you might not really need very often but which I still find worthwhile mentioning is the on-line encyclopedia of integer sequences. If you ever come across a sequence like 1, 1, 2, 4, 9, 21, … and you wonder what it is, this is where you find your answer.
MathOverflow is a truly amazing platform. It is a research level Q&A website with currently around 30k questions, 84% of which have been answered – many of them within one day. The platform is based on a reward system which is also used for StackOverflow (a Q&A site for computer programming with currently almost three million questions) and math.stackexchange which is made for/by people studying mathematics at any level. Both MathOverflow and math.stackexchange may be seen as collaborative mathematics blogs. By now, due to the impressive number of posts, they can also be used as encyclopedic resources: if you have a small question, it becomes more and more likely that it has been asked (and answered) there before.
A more concentrated effort towards solving specific mathematical research problems within an online collaboration is undertaken in the polymath projects. The speed at which those collaborations work is amazing (and at times frustrating when you want to catch up). I’m not even sure whether the product of time and manpower is an invariant of the problem here …
ArXiv and review databases
The easiest way to make a paper in mathematics, physics or computer science ‘public’ in the sense of making it ‘freely accessible to everybody’ is to post it on the arXiv. Fortunately, this has become the norm for a majority of scientists in those fields and the arXiv is now counting more than 748k publications. However, if you look for a particular result or for more recent papers citing a given one, the search options of the arXiv may not suffice. In such a case Google Scholar might help or, if you have access to those review data bases, MathSciNet and the ZBMATH online database. The latter contain reviews, summaries, citation lists and subject classifications. Very useful – and it would be even more so, if it did not require a subscription …
There is an enormous number of lecture notes out there – many of them really good, but unfortunately spread out and in this way hidden. One of the attempts to collect this material, at least locally, is MIT OpenCourseWare. There you can find hundreds (>2k) of courses taught at MIT in various disciplines. Some of them contain video recordings and exercises. A nice collection of interactive learning tools for mathematics is provided at TUM in the form of Mathe Vital.
I also appreciate more and more videos of lectures given at workshops, conferences or seminars. They have some obvious advantages compared to the real space experience: you don’t have to travel, you can schedule them whenever it suits you and you can skip, fast-forward and repeat. Clearly, there are disadvantages as well: you cannot interact and there may be neither free coffee nor cookies afterwards. A general platform where video lectures can be found is videolectures.net. As the domain name suggests, this is designed for exactly that purpose and I find it well done. In addition to the video, you are provided with a table of contents and a window containing the slides – everything synchronized.