I am applying for the NSF Mathematical Sciences Postdoctoral Research Fellowships, and I noticed the guidelines say the following:

The Project Description should be written with both specialists and non-specialists in mind.

I am not sure what exactly "specialist" and "non-specialist" refer to. Suppose that in my application, my proposed research objective is to solve the Navier-Stokes equation. (This is just an example!) Will all the reviewers be experts in PDEs? Or at least in analysis? Or is the most I can assume that they are mathematicians (possibly working in areas like algebra or topology)?

  • 3
    If your application was only going to be looked at by specialists in your field then you wouldn't be asked to make sure that your project description was written to be read by non-specialists as well as specialists. Sep 16 '19 at 22:21
  • 1
    There is a 2014 blog post from an experienced (pure) mathematician which is more about the standard NSF Research Grants than the PDRFs, but which might nonetheless have some useful details galoisrepresentations.com/2014/09/26/applying-for-an-nsf-grant
    – Yemon Choi
    Sep 17 '19 at 2:17

Some NSF programs are reviewed across sub-disciplines with reviewers that present a cross-section of mathematics. If memory serves me right, then that's true for CAREER proposals, for example, and I think also for the postdoc fellowships. In those cases, you'll have to expect that your audience consists of people in applied and pure mathematics, and you will have to find a level that keeps both of these groups interested in your proposal.

  • I suspect that the biggest communication gap is not best described as "pure versus applied", as though these were "two groups"... Further, not everyone self-describes as belonging to one of these two groups, after all. Sep 17 '19 at 0:18
  • @paulgarrett: But it is true that these two groups have rather different styles of communicating. Of course not everyone identifies with one group or the other, but largely, mathematics as a whole can still be divided into roughly these two camps. Sep 17 '19 at 15:17

I've never been involved in the reviewing process for NSF postdocs in math, but received one in 2012 and thus can say something about the way that I wrote my Project Description.

Looking back at my Project Description, I can see that the first page and a half was a general introduction to some of the theorems that I proved in my thesis / some papers I wrote in grad school. To give you an idea of what this means, the results in question concern spectral geometry, so I mentioned extremely famous problems and results like Kac's "Can you hear the shape of a drum?" problem, Milnor's examples of isospectral tori, etc. As I got closer to my own results things of course got more technical, but much of this section was written so that someone working in the general area of Riemannian Geometry would be able to follow the main thread without getting too lost. Basically, I wrote this section the same way that I'd write the introduction to a paper. (Try to motivate and describe the underlying ideas of my paper in as friendly a manner as possible, introduce technical notation only if necessary or if it is so standard that it won't limit my audience to specialists, etc).

The next half a page mentioned a few problems that built on my previous work and which I wanted to work on during the postdoc.

The third and fourth pages were more technical, but I still tried hard to make them accessible. One way to think about it is as follows. Suppose that you're going to speak in an analysis (or whatever big area you're interested in) seminar at a university other than your own. You're going to describe your work during the talk, so it's inevitable that you're going to have to go into technicalities that much of your audience isn't comfortable with. Still though, you want to keep things as friendly as possible for as long as possible. So you'll probably rely on special cases, analogies, heuristics, etc in order to convey the underlying ideas without having to introduce too many technicalities that are outside of your audience's comfort zone. This is the way that I wrote pages 3 and 4.

On the fifth page I mostly described why I would benefit from working with my sponsoring scientist and why the institution would be a great place to do math at.

To summarize, if I were in your situation I would write my introduction for a general audience of say, analysts. If you write for an audience much broader than this it's going to be tough to actually provide details explaining why your research merits NSF funding. If you want to aim some of the middle and later parts of your description for experts in PDEs I would think that's likely fine, but you should still try to be friendly enough that analysts not working in PDEs won't get lost and tune out immediately. For instance, even non-experts who get lost from time to time reading the description should still be able to retain the big picture (though perhaps in a fuzzy way).

Added: As I mentioned above, I have never been involved in the reviewing process for NSF postdocs in math. But my project description was definitely written in such a way that I can't see people working in totally different areas of pure mathematics (e.g., combinatorics) being able to get much out of it beyond the first couple of introductory paragraphs. Whether or not that was really an issue and I just got very lucky I can't say.

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