The Shapiro-Shapiro conjecture states the following. Let f : P^1 to P^n be any map. If all inflection points of the map (roots of the Wronskian of f) are all real, then the map itself can, after change of coordinates, be defined over R with real polynomials.

An equivalent statement is that certain real Schubert varieties in the Grassmannian intersect transversely -- a fact with broad combinatorial and topological consequences. The conjecture, made in the 90s, was proven by Mukhin-Tarasov-Varchenko in '05/'09 using methods from quantum mechanics.

I will present a proof of a generalization of the Shapiro-Shapiro conjecture, allowing the Wronskian to have complex conjugate pairs of roots. We decompose the real Schubert cell according to the number of such roots, and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is an evaluation of a symmetric group character. In the case where all roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture.

This is joint work with Kevin Purbhoo. Speaker(s): Jake Levinson (University of Washington)