Wednesday, August 17, 2011

Nima at PI

Nima Arkani-Hamed speaks at Perimeter this week, with an update on Grassmannian polytopes for scattering amplitudes. There is still no mention of associahedra or permutohedra, but they have been studying the work of Postnikov. The octahedron makes an appearance, complete with labels of four points in (sometimes degenerate) configurations on a line. Yes, in the removal of square faces for the permutohedron for $S_{4}$ one uses four point configurations. As usual, these labels are obtained from the areas between leaves on $5$-leaved binary rooted trees with ordered nodes.

Most of the talk focuses on positivity (strictly positive matrix minors) for Grassmannians. Recall that positivity leads to the study of tropical geometry, which is categorically nice because it introduces an arithmetic naturally associated to symmetric monoidal structure. It also leads to M theory brane diagrams with trivalent lines.

Nima concludes with a note that the non planar setting should be even more interesting, since similar combinatorial structures occur in other four dimensional theories.

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