  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.
