Abstracts for the Seminar

Fall 2016

**Speaker: **Steven Karp, UC Berkeley

**Title: **The m=1 amplituhedron and cyclic hyperplane arrangements

**Time:** 2:30 PM, Monday, October 17, 2016

**Place:** Malott 206

**Abstract:** The totally nonnegative part of the Grassmannian Gr(k,n) is the set of k-dimensional subspaces of R^n whose Plucker coordinates are all nonnegative. The amplituhedron is the image in Gr(k,k+m) of the totally nonnegative part of Gr(k,n), through a (k+m) x n matrix with positive maximal minors. It was introduced in 2013 by Arkani-Hamed and Trnka in their study of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. Taking an orthogonal point of view, we give a description of the amplituhedron in terms of sign variation. We then use this perspective to study the case m=1, giving a cell decomposition of the m=1 amplituhedron and showing that we can identify it with the complex of bounded faces of a cyclic hyperplane arrangement. It follows that the m=1 amplituhedron is homeomorphic to a ball. This is joint work with Lauren Williams.

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