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

**Abstract:**

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*: *CP*^{1} → *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 real spectrum.