Positive Genera of Amplituhedra

By recent work of Brown and Dupont, positive geometries can be understood via mixed Hodge theory, in which they arise from genus zero pairs. We show that this new framework is consistent with the known results for the amplituhedron. However, we also show that most amplituhedra do not make genus zero pairs. We then give an example of a positive genus pair, that still is a positive geometry, showing that the framework of Brown and Dupont is not equivalent to the original definition, and thus salvaging hope that amplituhedra are positive geometries. Based on joint work with Dmitrii Pavlov and Rainer Sinn.