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We prove the B. and M. Shapiro conjecture that says that if the Wronskian ofa set of polynomials has real roots only, then the complex span of this set ofpolynomials has a basis consisting of polynomials with real coefficients. Thisin 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 keyobservation is that a symmetric linear operator on a Euclidean space has a realspectrum.  In Appendix we discuss properties of differential operators associated withBethe vectors in the Gaudin model and, in particular, prove a conditionalstatement: we deduce the transversality of certain Schubert cycles in aGrassmannian from the simplicity of the spectrum of the Gaudin Hamiltonians.

The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz