The B. and M. Shapiro conjecture in real algebraic geometry and the   Bethe ansatz
  We prove the B. and M. Shapiro conjecture that says that 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 in particular implies the following result:   If all ramification points of a parametrized rational curve   $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset CP^r $.   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 a real spectrum.   In Appendix we discuss properties of differential operators associated with Bethe vectors in the Gaudin model and, in particular, prove a conditional statement: we deduce the transversality of certain Schubert cycles in a Grassmannian from the simplicity of the spectrum of the Gaudin Hamiltonians. 
E. Mukhin, V. Tarasov, and A. Varchenko