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As anyone who has ever applied to the NSF for a grant knows, such a proposal is a slightly odd piece of writing, not quite like anything else mathematicians are called upon to write. As such, it's a hard thing to learn to do well; of course, the basic requirements are set out in the grant proposal guide, and explained a bit more clearly in some other sources. But there's no substitute for reading an actual specimen.

Now, generally this is achieved by asking some older colleague to see an old proposal, an approach that works just fine in most cases. On the other hand, if there are any publicly available old NSF proposals online, I think finding them would be a boon to all of us preparing proposals, if only to have more data points. Does any one know of any (I didn't have much luck on Google).

I'll just note: I'm aware that the NSF has abstracts of all funded proposals on their website, which is useful, but doesn't give much insight into, say, what people write about broader impacts.

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If you're interested in broader impact in particular, you could write a separate question asking what kind of broader impacts people propose. – Emerton Sep 18 '10 at 5:33
I'd really encourage you to ask that separate question, Ben. If you don't, I might <grin>. – Andy Putman Sep 18 '10 at 15:08
Done. – Ben Webster Sep 18 '10 at 20:14

See Doron Zeilberger's page where he provides the proposals for several of his (approved) grants and the reviews (as noticed by Ben in his comment below) for two of them:

For other examples, see

SCREMS: The Computational Frontiers of Number Theory, Representation Theory, and Mathematical Physics

(it was awarded: see here)

and the awarded grant proposals by James Propp here and here.

By the way, this presentation also could be of some use:

Writing an NSF Proposal: a PI's and a panelist's perspective

as well as the Notices article

NSF Proposal Preparation: The View of an Ex-Program Officer

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Interestingly, he also includes the reviews of the grants (which at least for the most recent grant mention that the proposal is badly written, but the research is good enough that the reviewers don't care). – Ben Webster Sep 18 '10 at 0:11
Thanks, Ben! I've edited the answer to mention this. – mathphysicist Sep 18 '10 at 0:29

Off the top of my head: I think many recipients tend to put their proposals on their websites.

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William Stein places all of his grant proposals online here :

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I was just about to post that I do this, but you beat me to it (a more "stable" link is I also put some old job applications here: "Now, generally this [reading proposals] is achieved by asking some older colleague to see an old proposal, an approach that works just fine in most cases." There's one other way: if you ever get a chance to be on an NSF review panel, you'll also get to read lots of proposals. If you get a chance to serve in this capacity, take it no matter what! – William Stein Sep 18 '10 at 4:22

By law NSF must make the abstracts of all funded proposals public. They have a search tool for exploring awards: link. You can look for every proposal that includes, say, knot in its title (there are 29 active awards, not all in mathematics!). Etc. [Sorry to repeat what you knew, as indicated in your final note.]

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