A certain stratification of the Grassmannian of k-planes has recently turned up in several strikingly different contexts: Poisson geometry, nonnegative-real geometry, and characteristic p geometry. I'll focus on the last of these, explain the theory of Frobenius splittings, and show that the strata are naturally indexed by juggling patterns. There will of course be demonstrations.
Juggling patterns have previously been related to the Bruhat order on the affine flag manifold. I'll show how to derive the "cyclic Bruhat order" on the Grassmannian from affine Bruhat order. This will require introduction of antimatter (without demonstrations).
This work is joint with Thomas Lam and David Speyer.