The Poincaré Conjecture 

 While I'm sure that many readers of this blog saw   this article in the August 15 NYT science section , it's worth noting anyway as an insight into the sociology of mathematics and a look at some interesting pure mathematics as well (plus the graphics therein were really cool).  

 It seems that Perelman solved a 102 year old mathematical question of huge importance but wants to have nothing to do with the affect on the field and the resulting acclaim (including an almost certain Fields Medal) since he's disappeared into the Russian forest.  Nonetheless, other mathematicians have taken on the task of writing up his results producing proofs in  three books that are now available online .  These are fascinating to read, even though much of the discussion is at the highest mathematical level, since some of the principles are familiar to us (Cauchy-Schwartz, gradients, Hessians, etc.) from routine work, but obviously appearing in wildly different contexts. 

 So here's the related question.  Suppose, like mathematics, we could list the "big" unsolved problems in political science.  What would this list look like?  Personally, I'd love to see such a thing.  Of course it is unclear whether we have a David Hilbert-like figure to say "As long as a branch of science offers an abundance of problems, so is it alive" and then to go on and identify the 23 most important unsolved problems ( the 1900 "Hilbert Challenge" :).   In this vein,  my list of unsolved problems would start with why does the  discipline cling to the bankrupt NHST and continue to worship "stars"?