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There's a place to look this up. The name of the service escapes me at the moment, but it's from the AMS. Anyway, you'd just look up the problem and see who has published about it recently.
@Harry: I'm not sure what service you're thinking of, unless this was an ironic way of referring to MathSciNet, but I really doubt any online list will make it clear who has had some thought on a problem a few years ago and moved on, and who is continually returning to it. The latter information tends to spread through the mathematical community by word of mouth, so MO might be able to do better.
@Sean I think you should ask about one question at a time. For example: "Which mathematicians are actively working on quadratic generation of toric ideals of matroid polytopes?" strikes me as a fine title for an MO question.
@David: Yes, but I wasn't trying to be ironic. I really forgot the name of the service =).
I'm inclined to agree with gilkalai here. I feel like such questions would be better asked through e-mail. I'm sure there are people who you suspect are working on the topic. Perhaps you could e-mail them and ask something like "Are you working on this topic, and if not, do you know who is?".
I agree with Noah, this is quite sensitive and hard to give an answer which does not annoy anyone. Also, if you wonder what is going on about specific topic, you can ask questions like: "What is the latest developments/good survey on conjecture X/topic Y", etc. Often a good survey paper would point to most people who are working on such topic (admittedly, this is may be not up-to-date).
I agree with most of you here, and I think Hailong presents a good solution. There is a real difference in asking about latest developments since that always means you are asking about published or at least announced results. Where as if you ask who is working on what it might bring about speculations about mathematicians who are not active here, and might be private about what they are working on.
I have a friend in Minsk who has a friend in Pinsk …
who's friend in Omsk has friend in Tomsk ...
I can't wait to hear who saw the bear!
I think that's my cue to follow Antigonus' lead and exit this discussion.
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