Projects for the Real Solutions Texas A&M Research Group

Shapiro Conjecture experimentation.

Outline: Peform large-scale computer experiments to
study versions of the Shapiro conjecture in its many
manifestations and generalizations. The overriding goal is
to create software for this that runs mostly autonomously,
allowing similar experiments to be run easily in the future.

Problems:

The secant flags Conjecture. (For grassmannians, and then type-A flag manifolds)
Shapiro's conjecture for the Lagrangian Grassmannian (type-C)

These both continue through the classical flag manifolds
with the eventual goal of the exceptional flag manifolds.

Discriminants for the Ordinary Shapiro Conjecture

Use interpolation to compute these objects for the ordinary
Shapiro conjecture, as well as for the type A flag manifolds.
The goal is to show some are sums of squares, or have other
remarkable representations.

Maple Multivariate Interpolation Code (Vandebril, Van Barel, Ruatta, Mourrain) | MultiInterpol.tar.gz | guide

Analogs of an Eisenbud and Harris Result

Prove the analog of the Eisenbud and Harris result for the
Drinfel'd Grassmannian. Maybe also for the Lagrangian
Grassmannian.

Dynamic Symmetric and Skew-Symmetric Feedback

Generalize the paper of Rosenthal, Helmke, and Wang to dynamic
symmetric feedback as well as skew-symmetric feedback.

Partial notes on skew-symmetric problem | skewsymmetric.pdf