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. I work in stable homotopy theory, and will be applying for both tenure track teaching jobs in the US and post-docs in both the US and in Europe (with the job market as it is, it seems prudent to cast a wide net), but probably not at an R1 school. 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 found such a problem, I'd probably solve it myself rather than hanging onto it as a possible REU project.
(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.