A topological proof of the Shapiro--Shapiro Conjecture -- Jake Levinson, Aug 7, 2020.

Abstract: Consider a rational curve, described by a map f : P^1 \to P^n. The Shapiro--Shapiro conjecture says the following: if all the inflection points of the curve (roots of the Wronskian of f) are real, then the curve itself is defined by real polynomials, up to change of coordinates. Equivalently, 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 generalization of the Shapiro--Shapiro conjecture, joint with Kevin Purbhoo, where we allow the Wronskian to have complex conjugate pairs of roots. We decompose the real Schubert cell according to the number of such roots and, under a certain orientation, we prove that the topological degree of the restricted Wronski map is given by a symmetric group character. In the case where all the 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.