I often feel that the deluge of mathematical publications, fueled by the ever-increasing number of researchers and mounting pressure to publish, threatens to overwhelm my ability to keep up with advances. I don't think this is peculiar to applied mathematics. No matter how adept you are at sifting the chaff and finding the most relevant work in your field, you won't possibly have time to read every paper that is germane to your research, let alone those of tangential interest that might provide new research avenues. For my part, although I take time to read new papers every week, I've resigned myself to the fact that I won't see more than the abstract of most of the papers I'd like to read, because I need to conduct new research, teach, write, and so forth.
Reading and digesting a mathematical paper takes time and concentration. Nevertheless, I find that perhaps 80% of the value I get out of reading most papers can be summed up in a paragraph or two that is easy to read and understand. We all have practice producing those terse paragraphs because we regularly referee papers and provide a concise summary for the editor. This summary includes things like "what's really new in this work" or "how this relates to previous work", as well as an evaluation of its merit. Unfortunately, those referee reports are kept secret and unavailable to our colleagues. I mentally create a similar report for most papers that I read in depth, although I don't usually write my evaluation down and I certainly don't send it to anyone. What if every reader of a paper had access to the summaries and evaluations made by all the other readers? I think we could all learn a lot more, a lot faster, about what our colleagues are accomplishing.
Recently, Fields medalist Timothy Gowers proposed an approach to accomplishing just that. The idea is to bring the functionality of StackOverflow to the arXiv, creating a place where everyone can publish and everyone can openly referee or comment. The StackOverflow system of reputation and up-/down-voting would be used to help the best papers and best comments float to the top. As Gowers admits, there are plenty of obstacles, but I'm hopeful that people with his level of clout in the mathematical community could really bring this to pass. His interest seems mostly based on issues with the current journal publication system, but I see it primarily as a way to "collaboratively read" the literature. Indeed, it might be best if the site had no implications for decisions on hiring or tenure, to avoid any motivation to game the system. The site would also be a great place for expository writing that can't be published in a journal.
It's encouraging to see that some things are already moving in this direction. A new website named PaperCritic has just been launched to accomplish something roughly along these lines. It doesn't involve the StackOverflow system, but has Mendeley integration and allows you to post a public review of any paper. Meanwhile, an increasing number of scientists are including paper reviews in their blog posts -- something I would like to do here.
I think Mendeley could accomplish something useful in this direction if they would give users the option to make their library and notes public. Then when I find a paper on Mendeley that says "20 Readers", I could find out who they are, see what they've written about that paper, and see what else they're reading.
Note: I know that we already have Mathematical Reviews, but in my opinion it doesn't accomplish the goals mentioned above, mainly because the reviewer of a paper is often not sufficiently knowledgeable about the paper to say anything more insightful than what's in the abstract. I find that Mathematical Reviews gives me papers to review that I would never have read otherwise. What I'd like to see are reviews from the people who read the paper because it's germane to their own work.
I discovered while writing this post that there was until very recently a successful site of this kind used by quantum computing researchers called scirate.com. Perhaps we should focus on helping this guy get the site back up and start using it for math too.
Edit: Another brand-new open review system: http://open-review.org/
