I am completing a PhD in area of "applied" mathematics (although technically this general area is very broad and sits in between both pure and applied ends of the spectrum).
I am applying for some one-year post doctoral positions at another university. These positions are not attached to any specific member of academic staff, and are thus open to all applicants from pure / applied /statistics - a committee decides who gets the position, depending on a "number of various factors". I am meant to submit a research proposal, however. I have contacted an academic there whose research is in a similar area to mine (but still quite different, of course).
In my situation, given that I am still yet to complete my PhD thesis, and publish my results, what would be the best thing to do regarding the research proposal? How stringent are the requirements in general regarding research proposals for first-time postdocs in the eyes of professional mathematicians? I certainly have a number of potential research avenues to pursue stemming directly from my work - should I just write about them?