An REU is a "Research Experience for Undergraduates" and entails undergrads (usually after their junior year) coming for 8-10 weeks out of the summer to University X to learn about and work on a problem under the supervision of one of the faculty at X. They are often funded by the NSF and the students get a stipend. REUs seem to be very popular among small colleges because they want to be able to sell themselves as a place undergrads can do research even if there are no grad students around. As a grad student who will be on the job market in the not-too-distant future, I'm starting to wonder about how best to prepare and how to fit REUs into my job application material. I work in stable homotopy theory, and I have never come across an open problem that I think an undergrad could do meaningful work on (ignoring the small set of really amazing undergrads who frequent MO and would probably not end up at my hypothetical REU). If I did come across such a problem, I'd probably solve it myself rather than hang on to it as a potential REU problem.
A bit more background: I will be applying for a large number and variety of jobs because with the job market as it is, it seems prudent to cast a wide net. I'll be applying for post-docs in the US and in Europe, and I'll also be applying for tenure track jobs at schools which are more focused (but not entirely focused) on teaching. Long-term I hope to end up at a good liberal arts college, but in the short term I'll look for post-docs to strengthen my research profile. If I end up in a post-doc in Europe, I assume they will want me to focus on research and not running undergraduate research projects. Thus, this question is more focused on applications to colleges where teaching matters at least as much as research.
(1) should I spend some time seeking out a suitable REU problem in case a job asks me for project ideas?
I have no idea how common it is to ask a job applicant this type of question, but it seems like the sort of thing a small teaching college might want to know.
(2) Supposing that I did find a good problem and stopped myself from solving it, where would be the best place to display that in my job application material?
I've read up on teaching statements (on MO and elsewhere), but it seems that is not really the place to discuss plans for REUs. Similarly, it shouldn't go in the research statement. Is this just something I bring to the interview and hope they ask about?
(3) Do REU ideas need to be in the same field as my PhD research or could I just think up a good problem in a different field (e.g. graph theory where the background is more reasonable for an undergrad)
I'm amazed this hasn't been asked on MO before. I hope it's suitable, since many grad students frequent this site and would I suspect care about questions they may be asked in the job process. If it has been asked, please forgive the duplication. All I could find were questions about the merits of research if you are an undergrad, where to find info on REU type experiences, and the pros/cons for publishing with undergrads if you're looking to get tenure.