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