Date: Friday, September 17, 2021
Location: 4088 East Hall (3:00 PM to 4:00 PM)
Title: The m=2 amplituhedron and the hypersimplex
Abstract: I'll discuss a surprising parallel between certain decompositions of the amplituhedron, a nonpolytopal subset of a Grassmannian, and the hypersimplex, a polytope in R^n. The amplituhedron was introduced by physicists ArkaniHamed and Trnka to better understand scattering amplitudes in N=4 super YangMills theory. In particular, each "fine positroidal" decomposition of the amplituhedron conjecturally gives you a way to compute a scattering amplitude. The hypersimplex is a classical object in algebraic combinatorics; its decompositions correspond to tropical linear spaces (Speyer) and are parametrized by the Dressian. Despite the dissimilarities of the hypersimplex and the m=2 amplituhedron, LukowskiParisiWilliams conjectured a straightforward bijection between their fine positroidal decompositions. I'll discuss joint work with Matteo Parisi and Lauren Williams, in which we prove this bijection. Along the way, we prove an intrinsic description of the m=2 amplituhedron, originally conjectured by ArkaniHamedThomasTrnka; give a decomposition of the m=2 amplituhedron into chambers enumerated by the Eulerian numbers, in direct analogy with a triangulation of the hypersimplex; and find new cluster varieties in the Grassmannian.
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Speaker: Melissa ShermanBennet
Institution: University of Michigan
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