Experimentation and conjectures in the real Schubert calculus for flag   manifolds
  The Shapiro conjecture in the real Schubert calculus, while likely true for Grassmannians, fails to hold for flag manifolds, but in a very interesting way. We give a refinement of the Shapiro conjecture for the flag manifold and present massive (15.76 gigahertz-years) computational experimentation in support of this refined conjecture.   We also prove the conjecture in some special cases using discriminants and establish relationships between different cases of the conjecture. 
James Ruffo, Yuval Sivan, Evgenia Soprunova, Frank Sottile