AI-related honours projects?

I’m starting my Honours next year, and would like to do something towards helping MIRI with Friendly AI. I would also prefer to avoid duplicating any of MIRI’s work (either already done, or needed to be done before my honours are finished midway through 2015). I decided to post this here rather than directly email MIRI as I guessed a list of potential projects would probably be useful for others as well (in fact, I was sure such a thing had already been posted, but I was unable to find it if it did in fact exist). So: what sort of Friendly AI related projects are there that could potentially be done by one person in a year of work? (I suppose it would make sense to include PhD-length suggestions here as well).

Some notes about me and my abilities: I am reasonably good with math, though my understanding of probability, model theory and provability logic are lacking (I will have a few months before hand that I plan to use to try and learn whatever maths I will need that I don’t already have). I am a competent Haskell programmer, and (besides AI) I am interested in dependent type systems, total languages, and similar methods of proving certain program errors cannot occur, although I would have to do some background research to learn more of the state of the art in that field. I would (hesitantly) guess that this would be the best avenue for something that a single person could do that might be useful, but I’m not sure how useful it would be.