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I apologize if this has been covered before. A cursory search of meta.mathoverflow does not reveal it.

There are reasons not to make research topics public. (I have some topics that I am not ready to have the world discuss, for personal reasons.) Assume we have a Polymath spirit in this culture, and that none of the reasons apply. Is it appropriate to get an opinion from the MathOverflow community on specific research statement S?

I provide a couple of examples below to give an idea of specificity. I would also welcome examples from others as to how to frame such questions, as well as reasons why MathOverflow is not a good place for this. (I think it is a good place though.)

Example 1: In the spirit of Gowers's FUNC project, I decide to build a lattice of lattices: for each union closed family F on a base set of n-elements, I look at it in the collection of the subset of 2^(2^n) of union closed sets, and make a lattice L out of the collection (not necessarily a sub lattice of 2^(2^n) ). While my motivation is to search for counterexamples inside L, it may be that L itself will yield a counterexample for n large enough. Regardless of the motivation, what interesting questions can I find to ask about L? Would asking if L were modular be important? Can I make a good research question from it?

Example 2: Hearing people talk about the Collatz function turns me off; I will talk about the Syracuse problem instead. Suppose I want to consider dynamics of the problem over the Gaussian integers, where I introduce specific variant G of the Syracuse problem over Z[i]. Is this reasonable, or should I instead consider related problems over a class of algebraic structures?

Note that while many questions on MathOverflow have statements S and ask for references about S or proofs or refutations of S, or even relations of S to other statements T, the two examples above are a different type altogether. Although proofs, references, and related information about S are welcome, what is really desired are two things: an assessment of S as a worthy topic of study, and a family SS of related statements which might aid in the study of S or are preferred over S to be studied and of interest to some significant part of the professional mathematical community. Just as important, answering the question should not take much effort. If it is old hat, then a reference showing where it has been done is all that is needed. If it is poorly organized, there should be a quickly explainable reason. If it is of interest (and the reasons for not sharing don't apply), why not share it? If there is something easy that can be done to improve it, mention that in an answer.

Motivation: I recently asserted privately that a certain statement S would be worthy of proof/refutation, and am having second thoughts. It would be nice to get professional advice about lines of study. It would be too domain specific to ask about S on academia.stackexchange.com, and math.stackexchange.com also strikes me as the wrong forum.

Gerhard "Can We Crowdsource Graduate Advising?" Paseman, 2016.01.26

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    $\begingroup$ I didn't vote on this question, but I had to read it several times; I found it hard to read (i.e., understand). A question: are these examples made up, or are they supposed to be mathematically sensible based on real thinking (e.g., yours)? I think a status quo reaction, if these types of questions were actually posed, might well be "unclear what you're asking". Another phrase that comes to mind, used sometimes now but more often before the SE migration, is "fishing expedition" -- i.e. the OP has only a vague idea what he or she is after. $\endgroup$ – Todd Trimble Jan 27 '16 at 6:22
  • $\begingroup$ The two examples were improvised on the spot, so I expect them to be examples of (non-)specificity in some regards. In some way, the type of question does resemble a fishing expedition when it hopes for an SS answer rather than an answer to SS. A stricter measure would be, given introductory material I and brief research proposal P, I might write a research grant application including I and P. I might beforehand come to this forum and ask "Does P have a decent chance of being funded?" (continued) Gerhard "Wishing For Longer Comment Boxes" Paseman, 2016.01.27. $\endgroup$ – Gerhard Paseman Jan 27 '16 at 16:23
  • $\begingroup$ However, applying for a grant application is one of the reasons not to go public with P, so I don't come to MathOverflow to ask about P. Instead, I note that proposal P includes studying statement S, which may be slightly less specific than a normal MathOverflow question Q, but has a narrow scope as exemplified in my post above. However, I am unable to determine the suitability of S as a community accepted object of study. "Is S a good research question?" seems something that people here in the area of S might answer. Gerhard "Not Finished With This Yet" Paseman, 2016.01.27. $\endgroup$ – Gerhard Paseman Jan 27 '16 at 16:30
  • $\begingroup$ I am surprised that you might find it hard to understand. I don't consider the two examples above as good statements S that form part of research proposal P. I do consider them as examples of narrow scope: in one I am trying a specific route toward solving an open problem, but the object I pick may have its own intrinsic value for study. (Or does it? Whom should I ask?) The other is trying to generalize an open problem by changing domain. (Or did I generalize enough? Whom to ask?) Maybe I shouldn't have tried explaining the title. Gerhard "My Advisor Has Gone Fishing" Paseman, 2016.01.27. $\endgroup$ – Gerhard Paseman Jan 27 '16 at 16:37
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    $\begingroup$ Gerhard: what can I say? (Maybe I'm just not very bright?) Anyway, your last comment is a little confusing, since you say "I" but I think you mean instead some hypothetical poster. If that is the case, then the last instance of "I" seems to be one where you're switching back to the real "I" that is Gerhard Paseman. $\endgroup$ – Todd Trimble Jan 27 '16 at 16:56
  • $\begingroup$ I don't intend to express anything about your abilities. (I apologize for any offense caused.) I do intend to express my surprise about the question being hard to understand. Perhaps it is a communication issue. Regarding use of "I" in a previous comment, it does not matter to me if "I" always refers to me in that comment. Whatever attribution of people to "I" makes the most sense to you, use that. Regarding what you can say, I will post an answer later addressing that. Gerhard "Narrow Margins Serve Useful Functions" Paseman, 2016.0.27. $\endgroup$ – Gerhard Paseman Jan 27 '16 at 17:07
  • $\begingroup$ We seem to be getting off on a wrong track. Let me try another. Is this a type of question that you yourself are considering asking? Or, is this a type of question that someone you actually know might ask or might benefit from asking? Or is this really hypothetical, where you are imagining someone might ask such a thing in the future, and you'd like to know how the community should usefully respond? Etc. $\endgroup$ – Todd Trimble Jan 27 '16 at 17:22
  • $\begingroup$ I am considering the expansion of the purpose of MathOverflow (to judge as well as answer questions). While I myself would like to ask "Is S of current interest to (some part of) this professional community?" and get a yes/no answer, I think we could set up a format where others could say "is S' a good research question", and MathOverflow members could say yes or no with useful commentary. I could see new faculty and graduate students being served by answers like " No, did that in (reference)" or " Yes ( and X said this so far)". Gerhard "For Community And For Myself" Paseman, 2016.01.27. $\endgroup$ – Gerhard Paseman Jan 27 '16 at 17:37
  • $\begingroup$ Okay, that helps some; thanks. Still, I find the examples you gave not too clear. Example 1 for instance is really vague, and it's hard to tell whether this is supposed to be a signal that the (let's say young) researcher doesn't want to reveal his hand, or it's because you don't really have a specific example but just wanted to give a general flavor of the type of question. [Because if it were really phrased like that, it would be entirely justified to vote to close right away as "unclear what you're asking".] I'm pretty confident guessing it's the latter, but (continued) $\endgroup$ – Todd Trimble Jan 27 '16 at 17:48
  • $\begingroup$ but I think it would probably help your post if you could think of a more plausible and specific example, something that would not automatically be shut down in five minutes. That's just my opinion. $\endgroup$ – Todd Trimble Jan 27 '16 at 17:49
  • $\begingroup$ Anyway, your question is interesting. I'm undecided if it's a good idea. $\endgroup$ – Todd Trimble Jan 27 '16 at 17:56
  • $\begingroup$ Thank you for your input. I am sure more appropriate examples exist, and if you understand the thrust, I would be appreciative if you could counter with better examples ( even if ultimately the answer to the title is "no" ). One good answer to this post would have a form like "There are already several references as to what makes a good research question, see A, B, and C. MathOverflow cannot do as good a job for the simple reason that R. Therefore no, it is not a good MathOverflow question." Gerhard "Sometimes Prefers Abbreviations Over Acronyms" Paseman, 2016.01.27. $\endgroup$ – Gerhard Paseman Jan 27 '16 at 18:17

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