In this talk we will see how the equivariant cohomology of the flag variety
is both strictly better, and strictly easier to compute than the ordinary
cohomology. GKM theory allows us to relatively easily present the
cohomology as a direct sum of polynomial rings. We will then see the
definition of the Peterson Variety, which is a subvariety of the the flag
variety. It's ordinary cohomology has only been computed by first
computing its equivariant cohomology. As with many of the cohomology rings
studied in Schubert Calculus, the equivariant cohomology of the Peterson
variety has an elegant combinatorial presentation. Giving this
presentation, for all Lie types, was a significant portion of my
dissertation.