Abstract: I will explain a proof of the BCFW triangulation conjecture which states that the cells appearing in the Britto–Cachazo–Feng–Witten (BCFW) recursion triangulate the amplituhedron (in full generality at all loop levels). The key ingredient is a relation to origami crease patterns which are planar graphs with faces colored black and white, embedded in the plane so that the sum of black (equivalently, white) angles at each vertex is 180°. Along the way, we prove conjectures of Chelkak–Laslier–Russkikh and Kenyon–Lam–Ramassamy–Russkikh on the existence of such origami embeddings of arbitrary planar graphs, which originated from the works of Kenyon and Smirnov on the conformal invariance of the dimer and Ising models.

Seminar talk is supported in part by the Mrs. Hepsa Ely Silliman Memorial Fund.

Abstract: I will explain the combinatorics behind the cluster algebra structure on braid and Richardson varieties. This generalizes the previously known constructions for positroid varieties and double Bruhat cells. The cluster algebra is described in terms of a 3-dimensional generalization of Postnikov’s plabic graphs, and the underlying quiver is induced by the intersection form of the Goncharov–Kenyon conjugate surface associated to a 3D plabic graph. Joint work with T. Lam, M. Sherman-Bennett, and D. Speyer.