How does one find out what's happening in contemporary mathematics research?
EDIT: I should have mentioned that I am looking for open access online sources. It so happens that I have been outside academia for quite a few years, and the gap between grad students' access to recent research (which is in the air, word of mouth + all publications immediately available) and that of a graduate who now works in an industry (few contacts if any + no access to journals) is tremendous. IMO this gap is the primary reason it's so difficult to do mathematics "on the side".
There are a few personal blogs out there, but personal blogs, even as very good ones, cannot be all-encompassing and necessarily focus on the subfields that the blogger is interested in most. There is MO, of course, but this is a Q&A site rather than a newsroom. There is word-of-a-mouth from other mathematicians, but the breadth of news one learns that way depends too much on one's contacts.
Physicists have Physical Review Focus website, which provides very accessible overviews of both recent and significant developments in Physics. This is not somebody's personal blog, but a well-organized collection of short digests by many contributors with references to the original papers. On most of the recent significant developments one gets a high level overview written by a specialist for a non-specialist, and can get access to a more detailed paper by following the supplied references.
I wonder if there is a site like that for mathematicians. Please tell if you know of one. If not, perhaps one could be built based on MO resources?
Best Answer
The question may be understood in different ways, depending on the time scale (what do you mean by "recent"? last week? last year? last three decades?) and the level at which you wish the mathematical developments to be presented (for the general public? for beginning graduate students? for professional mathematicians in general? for specialists of the field?). So the question will not have any single good answer.
Also, and here it is perhaps a difference with physics, short time scale are not very relevant in mathematics. Most of the important new ideas takes years, often decades, to be checked, developed, known, understood. So it is not a big deal if you miss the first time (or the second the third, etc.) that an important development is announced: you'll have years to catch up, and after sufficient time the news will come from so many directions that you will not be able to miss it. This may explain why there is no equivalent of the Physical Review Focus in mathematics.
That being said, let me discuss the resources I know. At a very short time scale, the newsletter Headline and Deadline of an AMS announces surprising advances (e.g. the bounded gap between primes). Mathoverflow, even though it is not its aim, also does in practice. On a longer time scale (about 5 or 10 years), one famous resource is the Bourbaki Seminar, which tries to cover important development at a specialist level. Of course, it has never been perfect in covering all mathematics, and my opinion is that it is much less good in that respect that it used to be. Yet it is a valuable resource.