Will I be allowed to reapply for a mathematics NSF postdoc next year after declining one this year? I am a mathematics PhD student in the US graduating this year. I recently received a very good 3 year postdoc offer in Europe, which I have to accept or decline in early December. Not being a big risk taker, I am leaning toward accepting the offer.
I have also applied for an NSF postdoc, which I would prefer to the European offer but am obviously not a shoe in to get--and the results of which won't be known until February. If I accept the European offer, I will have to decline the NSF postdoc if awarded.

If I do that, would I be allowed to reapply for the NSF next year?
If you accept another position before the NSF Mathematical Sciences Postdoctoral Research Fellowships are awarded, you should officially withdraw your NSF application.  The way the NSF postdocs work is that there is no second round of offers: if you are awarded one and turn it down, then it won't be offered to their next choice of candidate, who might have received it if you had withdrawn your application.  Furthermore, I believe withdrawing will preserve your eligibility for next year, while you will not be eligible if you decline an offer (according to the official instructions).