In this question, the asker needs to choose a topic for an independent study project, and is asking for suggestions.
Are such questions appropriate here?
As the comments express some differing opinions, I thought I would start this meta question rather than casting the final close vote. (I've added my own opinion as an answer below.)
I agree with Nate Eldredge that such questions lack objective, definitive answers. Possibly they could exist as community wiki questions, although that format seems to be going out of fashion on SE sites.
More fundamentally though I think that students who ask for research topics on the internet are making a mistake, and probably not taking proper advantage of the resources available to them. Beginners at mathematical research need, and deserve, significant mentoring. It is hard for me to imagine an academic context in which research is asked of students (especially, undergraduate students) but no instructor in the department is willing to provide significant aid. I think it is much more likely that the students who are asking here for help are simply shy, or are misunderstanding the amount of independence that is expected of them. Certainly if I assigned a research project to an undergraduate and found out that she came here asking for such broad, initial help, I would feel that I had failed to make myself sufficiently available to the student.
My feeling is that this is not a good fit for the Q&A format. It doesn't have an objectively correct answer, and will likely result in everybody suggesting a project in his or her own favorite area. I would prefer that such questions be closed.
Personally, I'm okay with it. I agree with Pete Clark that math.SE might not be the optimal source of research topics, and that ideally a student will be guided by the course instructor.
But then again, math.SE may be the best resource available in some cases. I remember being assigned independent topics to present in high school geometry; if only math.SE were around at that time, I may well have gone and learned the basic definitions and a model for projective geometry, which I'm sure my teacher was unaware of.
Although questions of this type lack an objectively correct answer, the math.SE community may be the best source for interesting-yet-accessible topics. Sure, plenty of people will just post their personal favorites, but after a few comments asking about the student's background, I think voting will often push the most helpful suggestions to the top.
So I propose we leave these questions open, as community wiki.
I agree with Brett, we shouldn't assume that all students have access to top faculty or mentors. Furthermore, not all posters asking for topics are students in an institution. There are many interesting topics/problems that can be difficult to find yet can be done independently for pure enjoyment or personal study.
Perhaps such questions could be improved by narrowing down the subject area sufficiently. For example, asking about the active areas of research or examples of unsolved problems within a specific field could draw attention of the specialists and thus make responses more objective, rather than merely a reflection of personal preference.
