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Dear Joseph,
I'm not aware of MSE attracting a high number of research-level questions that don't originate from Akhil Mathew, Qiaochu Yuan, or a few other high-level undergraduates here who, as a sign of (false?) modesty, have decided to ask their questions there rather than here.
At least as far as those two are concerned, I think that if MO requests that they ask questions here rather than there, they would happily return to asking questions here.
In fact almost all the high-ranked (say top 20) contributors on Math.SE seem to also participate in MO, so there doesn't seem to be much danger so far of expertise being lost to one site or another.
MSE has now posted 23,993 questions, surpassing MO's 23,992 questions. I don't know the exact birthdates of each forum, but MO is roughly twice as old as MSE. Like the porridge, I find MO's question-rate, which I guess to be 40 questions per day, just about right. MSE's at about 100 questions per day is too hot for my tastes, and CSTheory's 10(?) per day too cool. :-)
Not being active on MSE (and only looking there on occassion) I am neither well placed to judge this in detail.
However, it is from an MO persepective my opinion that a certain type of question that in my opinion used to be more present on MO, and I assume this is precisely the type that now might be asked on MSE, starts to be lacking on MO. What I mean are perhaps too vaguely described "solid graduate level questions" (not the most standard textbook-excercise question from some graduate Analysis or Algebra course but, quoting the FAQs emphasize mine, "the sorts of questions you come across when you're writing or reading articles or graduate level books"; in other words question that are a bit beyond what one can find in any book on the subject but questions that might come up when thinking beyond what one read).
I believe that, ultimately, it is the smaller number of this type of questions that causes the at first glance paradoxical situation that on the one hand some people say that the level of MO significantly increased while at the same time I see some (at the strict/high-level end) users say that MO used to be better.
I think quid's comment is exactly right.
Echoing quid: when we started MO, I know I asked a number of (naive) graduate-level questions. I am, of course, still "graduate level", but the questions I've been thinking about more recently are more focused, and either things I would like to think about privately, or things that I know the one expert to email and ask. I do hope that MO has (or continues to have) "graduate level" questions; for example, questions of the form "this came up in my class on ..., and the professor didn't know the answer off the top of her head" are, I think, very appropriate.
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