Experimentation and conjectures in the real Schubert calculus for flag   manifolds (extended abstract)
  The Shapiro conjecture in the real Schubert calculus fails to hold for flag manifolds, but in a very interesting way. In this extended abstract, we give a refinement of that conjecture for the flag manifold and present massive experimentation (over 12 GigaHertz-Years) that supports our refined conjecture. We also establish relationships between different cases of the conjecture and describe some new phenomena uncovered in this experimentation. 
James Ruffo, Yuval Sivan, Evgenia Soprunova, and Frank Sottile