TITLE: How to know if somebody else is also working on your problem?
QUESTION [50 upvotes]: Once you have spotted a mathematical problem that (presumably) fits your degree of expertise, whether you are a phd student or an established professor, you have to deal with the following non mathematical problems:

How to know if somebody else in the world is already working (or has already been working) on the same problem? 

If the other guy has already completed a certain amount of (say, not yet published) work on that specific topic, knowing this would help you to avoid waisting time to try to re-do something that has already been done (at least with the same methods).
On the other hand, if the problem is broad enough, knowing of some other's interest in the same context would also be fruitful because you'd may have somebody with whom to talk and to whom to ask questions, without overlapping the specific research goals. Or you may even find a collaborator.

In some cases the very choice of an interesting specific problem can be a nontrivial task by itself. So, in case you want to ask around if some previous/present effort has/is been made in that specific direction or related ones, should you worry about the possibility that somebody with a higher degree of expertise would just "take your problem" and solve it faster than you would do? 

I'd expect the obvious answers, such as "have a look to mathscinet/arxiv" or "search the literature" or "talk to people (your advisor if you're phd)", to be enriched -if possible- by some more elaborate viewpoint or more specific suggestion.

REPLY [21 votes]: Recently I'd started working on a project with a leading expert in the relevant field where the result we'd worked on turned out to be already in the literature unbeknownst to either of us.  So this can happen even to leading experts.  (Our method of proof was slightly different, but that part of the project instead of being a new result just becomes the pushout of a known result with a known technique proving a similar result, and so not really worth publishing.  (Though it was still a project where I learned valuable things!)  Fortunately we hadn't actually started writing.)
So how did we figure out that this result was in the literature?  One day I decided to read the mathscinet reviews of every paper with primary subject classification 46L37.  The reason I did this was actually to try to learn where subfactor papers were published, but a nice side affect was that I learned about this prior excellent work so we didn't put in a bunch of effort only to discover that someone else had done it several years ago!
I highly highly recommend at least skimming through the mathscinet listings of subject classifications that you tend to publish in.  It really gives a nice birds eye view of the field (albeit a couple years out of date).  (Of course, this is easier to do in fields that have only existed for 30 years.)

REPLY [11 votes]: I'm surprised that no one has posted the following response yet:
Post a question on MO asking what is known about your problem.
I'm curious to hear people's opinions about this.  Is this within the intended purpose of MO?  In a sense, this is certainly a "research-level question".  Some people may be more cautious about broadcasting what they're working on, so as to avoid attracting others to the problem.  But we certainly have a large community of active mathematicians here, and one might gain a lot of insight from the answers to such a question.