Nima Arkani-Hamed of the Institute for Advanced Study continues his lecture on scattering amplitudes and the dualities that emerge when these calculations are pushed to infinity, delivered as part of the Prospects in Theoretical Physics 2026 program. This second installment builds directly on groundwork laid in part one, working through the mathematical structures that relate different formulations of particle scattering processes and the geometric objects, such as positive geometries and polytopes, that encode them. Arkani-Hamed works at the blackboard through the technical machinery connecting amplitudes to deeper combinatorial and geometric principles, the kind of material aimed at researchers and advanced graduate students already fluent in quantum field theory. The talk sits within his broader program of recasting particle physics calculations in terms of underlying mathematical structures rather than traditional Feynman diagrams, and it assumes the audience is tracking a conversation that began in the first session.