Shapiro Conjecture : Part I

Part I: The Wronski Map and Grassmannians

The simplest statement of the Shapiro conjecture concerns the Wronski map.

While the original Shapiro Conjecture was much more general, the version which has been most studied (and which is not known to be false) concerns the Schubert calculus on the Grassmannian.

Part I treats this case, depreciating the Schubert calculus to make the ideas accessible.

Part II will discuss a counterexample to the conjecture for the flag manifold, and two generalizations for which there is some evidence.

There is likely much more to this story, but even the statement of the conjecture is not clear for other flag manifolds.