Boris Shapiro and Michael Shapiro have a conjecture
concerning the Schubert calculus and real enumerative geometry and
which would give infinitely many families of
zero-dimensional systems of real polynomials (including families of
overdetermined systems)---all of whose solutions are real.
It has connections to the pole placement problem in linear
systems theory and to totally positive matrices.
We give compelling computational evidence for its validity, prove it
for infinitely many families of enumerative problems, show how a simple
version implies more
general versions, and present a counterexample to a
general version of their conjecture.