We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.
García-Puente, Luis D.; Hein, Nickolas; Hillar, Christopher; del Campo, Abraham Martín; Ruffo, James; Sottile, Frank; and Teitler, Zach. (2012). "The Secant Conjecture in the Real Schubert Calculus". Experimental Mathematics, 21(3), 252-265.