  We aim in this manuscript to describe a specific notion of geometric
positivity that manifests in cohomology rings associated to the flag variety
$G/B$ and, in some cases, to subvarieties of $G/B$. We offer an exposition on
the the well-known geometric basis of the homology of $G/B$ provided by
Schubert varieties, whose dual basis in cohomology has nonnegative structure
constants. In recent work [22] we showed that the equivariant cohomology of
Peterson varieties satisfies a positivity phenomenon similar to that for
Schubert calculus for $G/B$. Here we explain how this positivity extends to
this particular nilpotent Hessenberg variety, and offer some open questions
about the ingredients for extending positivity results to other Hessenberg
varieties.
