Positive Geometry Beyond Integrands

Since the discovery of the Amplituhedron in 2013, positive geometry has attracted considerable interest in both physics and mathematics, revealing connections between scattering amplitudes and cosmological correlators on the one hand, and combinatorics and algebraic geometry on the other. I will review the relevant ideas in the context of planar maximally supersymmetric Yang--Mills theory, where the Amplituhedron geometry encodes tree-level amplitudes and loop-level integrands. I will then emphasize recent developments aimed at understanding the role of positive geometry beyond the integrand, after loop integration. The broader goal is to develop geometric methods capable of accessing information about amplitudes that is difficult to obtain through more standard techniques.