How should one interpret a requirement that a proposal in pure math be "jargon-free?" I'm applying for a fellowship wherein they ask for a long proposal about what I'm working on (a PhD thesis in pure math).  Then they advise me that some of those evaluating me will be from non-math departments, so the proposal "should be jargon-free."

Right off the bat I guess [PSL2(Od):Γ] &lt; ∞ is out of the question...  I could explain what a manifold is to a mixed audience, but I think they'd need to do some homework before they followed what it might have to do with finite index subgroups of Bianchi groups (or whatever topic you might be studying).  Without the "jargon" I feel like my whole language for it is gone. Afterall the reason we have so many definitions is because each one refers to a distinctly defined thing that we previously had no name for!

I can see that I could take a more historical, conceptual perspective, but I'd still be dancing around what I'm actually doing.  It's especially hard to summarize something that is not even fully developed.  I think that for people who don't study math, the only reference point is science applications, but those don't really exist here because then it would be applied math and not pure math.

I'm coming up short looking for anyone addressing this online.  There are multidisciplinary tips about writing proposals, but I think the accessibility problem is at its biggest with pure math.
I think that the key idea here is that they want to know why you are doing what you are doing more than what you are doing.  

Let's say you're diligently working towards establishing an isomorphism between soliton elves and the christmas-tree singularity.  That's all well and good, but:


There are lots of properties in the world.  Why is isomorphism between soliton elves and the christmas-tree singularity an interesting property to try to establish?
Is there a higher-level strategy to your approach than "just do lots of math until the problem is solved"?
Why is the approach that you are taking a good one?
If you establish this property, what are the consequences, and what comes next?


A really good and clearly written proposal should be able to have its logic grasped by pretty much any scientist, whether or not they are capable of judging its originality, plausibility, and significance in detail.