In 1995 Boris Shapiro and Michael Shapiro made a remarkable conjecture concerning real solutions to enumerative geometric problems on the Grassmannian. Later, Sottile tested a few instances of these conjecture and much to his surprise found it to hold in every case. The apparent validity of this conjecture has inspired a significant amount of research, including a previous experimental project of Sottile that inspired this current project. While the general conjecture of Shapiro and Shapiro remains open (in 2003), it was established for Grassmannian of 2-planes in n-space by A. Eremenko and A. Gabrielov [EG00].
The original conjecture of Shapiro and Shapiro dealt not only with Grassmannians, but also with more general flag manifolds. (An excerpt from an email message of Autumn 1995 from Boris Shapiro to Sottile.) It is false in this wider generality. However, it fails in a very interesting way, and the purpose of this project is to investigate this failure and to make new conjectures concerning the classical manifold of flags in n-space. These new conjectures are generalizations of the Shapiro conjecture forGrassmannians.
Supporting these new conjectures are first and foremost massive experimental data. To date, we have used 15.76 GHz-years of computer time to investigate 1126 different enumerative problems on 29 different flag manifolds. In all, we have solved 525 420 135 polynomial systems; determining the numbers of real and of complex solutions in each. In addition, we are able to prove the conjectures in 0 cases by examining discriminants. We also have theoretical results linking different enumerative problems, this proves infinitely many instances of the conjectures, by reducing them to cases previously established. In short, we are quite confident of the validity of these conjectures.
The primary purpose of this document is to archive our computations and facilitate our data analysis. However, it will also provide context, with a description of the geometry of the flag manifolds and Schubert varieties, statements of the conjectures, and statements of some theorems we have established concerning them. This document contains analysis of the instances established by examining discriminants, as well as a discussion of how the data was taken. All software used in this project is also archived here.