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I am actually still in High School, but I enjoy some more advanced forms of mathematics. I asked a question, and it generated a great response, and I really enjoyed the process, so must have come of as research-level. On the other hand, I asked a question here, and I was politely notified it was not at the level of this site.

If I am not currently in academia, what criteria can I use to determine if a question is research-level?

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    $\begingroup$ Wow, you're in high school? Your responses seem very mature. I'm not sure about the "research level" though of your popular question -- it just happens sometimes that a question tickles the fancy of users. Your question here by the way is a tough one. $\endgroup$ – Todd Trimble Jan 27 '15 at 23:09
  • $\begingroup$ @ToddTrimble I am not always mature? $\endgroup$ – PyRulez Jan 27 '15 at 23:12
  • $\begingroup$ Part of it is recognising what is a grammatical statement in the language of research mathematics, another part of it is recognising that the statement is meaningful ("I ate grandma's fish scooter" is a grammatical sentence but is pretty meaningless), and a third part is being able to make some sort of Bayesian judgment that the question is a sensible thing to ask (for instance: "is my proof of Riemann using my proof of Goldbach correct?" is not sensible, because the answer, almost surely, will be no, with nothing learned by anyone). Each of these requires familiarity that comes from use/time. $\endgroup$ – David Roberts Jan 28 '15 at 1:14
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    $\begingroup$ "research level" is a code phrase for "acceptable to most of the MO community", and is a characteristic that changes over time. Learn by doing, and take closures and criticisms as feedback, and not as social censure. $\endgroup$ – The Masked Avenger Jan 28 '15 at 3:23
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    $\begingroup$ It may also be good to note that one cannot always judge if a question will be well-received based on previous questions if those questions are old enough (not that you are doing that here, it is just a general comment). I have a feeling my first question here would not have been received nearly as well today for example (nor possibly some of my other early questions). $\endgroup$ – Tobias Kildetoft Jan 28 '15 at 8:04
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    $\begingroup$ @PyRulez I should point out that I ask quite a few questions which are hard for me on math.stackexchange, just because I know that they are not "up to snuff" for MO. One can profitably use both sites for their respective purposes. $\endgroup$ – Steven Gubkin Jan 28 '15 at 17:37
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    $\begingroup$ @David: I'll have you know that grandma's fish scooter is one of my favorite dishes! :-) $\endgroup$ – Asaf Karagila Jan 28 '15 at 21:26
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The question is probably, "how can I tell if my own questions are 'research level', given the fact I am not an academic?"

I think a generally healthy attitude to have in this situation is that you should consider yourself pretty darned lucky (at this stage of your career) to hit on a question which could be considered "research level" (much less cutting edge!), and thus you should be a little bit skeptical towards your questions being "professional level" before you've gained more years of experience. I don't want to discourage you by saying this, but I do want to encourage a realistic attitude.

I feel even more confident saying this since your two questions have been about category theory, the area in which I mostly work. As you know category theory has been worked over by hundreds of very smart people over a period of about 70 years and concerns extremely general structures of essentially algebraic type. The thing about such generality is that your question -- especially if it concerns basic concepts such as cartesian closed categories -- is with great probability either one where people know how to construct counterexamples (because, after all, with great generality comes great flexibility), or if not, it's probably true for some natural reason that people have hit upon after 70 years of collective effort. (It's different from how it is with say number theory, where the integers are what they are and there is less built-in flexibility, so to speak -- there one can hit upon some question that no one knows how to answer with somewhat greater probability.)

Anyway, if you want to submit categorical questions to MO, you should definitely do a little research on them beforehand. Use Google, Wikipedia, the nLab, and of course MathOverflow and Mathematics StackExchange, just to name a few. If the structures you are asking about have been studied since the 60's -- and this is something you can probably find out through judicious use of Google -- the chances are high that answers have been around for a while. On the other hand, if you are talking about more esoteric matters such as differential cohomology in a cohesive $(\infty, 1)$-topos, then maybe not -- but there the bar is pretty high for knowing how even to formulate a sensible question. (I've seen some evidence of some young people overreaching in this respect, and it leaves a bad taste.)

I suppose that people might assign greater cachet or prestige to MO than to Math.SE, but you should consider using both, and use your judgment which is likely to be more appropriate. (Unfortunately, it is really M.SE that is "Overflowing" with questions these days, with many questions going unanswered, and not because they're "too hard" for M.SE.)

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    $\begingroup$ Oh, by the way: learn lots of mathematics! All category theory and nothing else makes Jack a dull boy. :-) $\endgroup$ – Todd Trimble Jan 28 '15 at 4:00
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    $\begingroup$ Yeah, I was thinking that first question may have been a fluke. (I wonder if we should have intermediateflow, for intermediate people.) I'll make sure to do more research on my questions (and learn more maths.) $\endgroup$ – PyRulez Jan 28 '15 at 11:30
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    $\begingroup$ Todd, I didn't know that Math.SE is "overflowing". That's a useful remark that perhaps should be disseminated in MO somehow. (I haven't used Math.SE to this point, but maybe I will in light of your remark.) $\endgroup$ – Nick Gill Jan 28 '15 at 14:09
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    $\begingroup$ @NickGill That's my perception anyway. Another problem I see personally is that many responses have a hard time getting attention or traction: sometimes when I go there to answer, my posts get no feedback (no votes, no comments, no nuttin'). If that happens to others as well, the feeling might be: what's the point, then? (But, it probably doesn't help that I identify there as userxyzuv -- silly, perhaps, but I had wanted to see how it felt being a semi-anonymous user.) $\endgroup$ – Todd Trimble Jan 28 '15 at 15:02
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    $\begingroup$ @PyRulez: For what it's worth, I'm a math Ph.D. student, and I've both asked and answered questions on Math.SE that were directly relevant to my research. To me, Math.SE is MathIntermediateflow, and I don't think there's any harm or shame in asking sophisticated questions there. $\endgroup$ – Vectornaut Feb 4 '15 at 7:39
  • $\begingroup$ @ToddTrimble, I strongly agree with your feeling. I'm also a high-school level ""mathematician"" and in the last 6 months I often had the (bad)idea to come here on MO because my questions did remain(and still are) unanswered for days and days, even with 2-3 consecutive bounties!!!. For few seconds I thought that was for the extreme sophistication and complexity of my last questions... well I'm joking...I guess that MSE is really getting less active. $\endgroup$ – MphLee Feb 16 '15 at 18:30
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    $\begingroup$ @MphLee Thanks for your feedback. I am beginning to believe that M.SE is becoming in many ways highly dysfunctional. I rarely have a good feeling when I am over there (frankly, I feel a great deal more camaraderie and mutual support here than I do there). I'm not sure how to get a good and frank discussion going about this. $\endgroup$ – Todd Trimble Feb 16 '15 at 18:52
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    $\begingroup$ @ToddTrimble I think that it could be a very complex problem to face imho. The first reason is that probably the perception of a drop in the efficiency could be seen as subjective... or better... local. continue $\endgroup$ – MphLee Feb 16 '15 at 19:04
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    $\begingroup$ continueI mean, I guess that MSE (as MO probably) is composed by alot of subcommunities(by topic) that seems to have a considerable degree of independence so is possible that the drop of "efficiency" or "functionality" happened to few subcommunities while other have remained the same, or improved. In other words, maybe only my favourites topics have less ativity, or something like that. Another reason is that I don't see how we could fix a lack of interest in some topic (set of tags), if this was the real cause. $\endgroup$ – MphLee Feb 16 '15 at 19:05
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    $\begingroup$ @MphLee Those are excellent points! Thanks again. $\endgroup$ – Todd Trimble Feb 16 '15 at 19:25

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