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Notice unusual time The bootstrap program aims to determine scattering amplitudes from nothing more than a few physical principles and a thorough understanding of their analytic structure. In the case of N=4 super-Yang-Mills theory, all information about the analytic structure is in principle encoded in the geometry of the amplituhedron. I will report on work in progress to use the amplituhedron to facilitate the analysis of Landau singularities of multi-loop amplitudes in this theory.