First research proposal for a postdoc application 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?
In mathematics, your situation is very common.  A lot of people don't have any papers written and are still completing their PhD when applying for their first postdoc.  There's nothing "stringent" about "research proposal requirements."  (Are there even any requirements?  Typically, just that something be submitted, I suppose.)  What you'll be evaluated on mostly at your stage is your area of research and letters of recommendation (and possibly your teaching ability, depending on the position).  Yes, people will look at your statement, and you should do a good job at writing it, but don't worry too much about it.

Generally what you should do in a research statement at your stage is:


summarize what you've been working on, and your (partial?) results till now
indicate, in some but not a lot of detail, what are some future things you are interested in working on (these do not need to be things you're definitely going to do)


The point of this statement is:


To see that you can effectively communicate your work/ideas (and possibly to get a sense of how well you understand its context)
To see if what you're doing is of interest to the faculty there
To see that you have some decently thought out ideas for future work