The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
I shall discuss the proof of Shapiro's conjecture by methods of math physics. The Shapiro's conjecture says the following. If the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients.
This statement, in particular, implies the following result: If all ramification points of a parametrized rational curve f: CP1 → CPr lie on a circle in the Riemann sphere CP1, then f maps this circle into a suitable real subspace RPr \subset CPr.
The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has real spectrum.