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 withâ€¦ (More)

We describe a large-scale computational experiment studying structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problemsâ€¦ (More)

Traditional formulations of geometric problems from the Schubert calculus, either in PlÃ¼cker coordinates or in local coordinates provided by Schubert cells, yield systems of polynomials that areâ€¦ (More)

The Monotone Secant Conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the Monotone Secantâ€¦ (More)

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here, we consider Schubert problemsâ€¦ (More)

Formulating a Schubert problem as the solutions to a system of equations in either PlÃ¼cker space or in the local coordinates of a Schubert cell typically involves more equations than variables. Weâ€¦ (More)

We establish a congruence modulo four in the real Schubert calculus on the Grassmannian of m-planes in 2m-space. This congruence holds for fibers of the Wronski map and a generalization to what weâ€¦ (More)