In this talk, I will give a gentle overview of recent developments at the intersection of combinatorics, real, complex and tropical geometry and theoretical particle physics. The beating heart of that intersection is the Cachazo-He-Yuan (CHY) integral formulation of Quantum Field Theory, and its generalization by Cachazo-Early-Guevara-Mizera (CEGM) to moduli spaces of points in projective spaces of any dimension. The values of these integrals are rational functions of certain kinematic invariants, closely related to a richly structured object, the tropical Grassmannian; when these invariants are taken to be non-generic, beautiful combinatorics emerges. We explore that combinatorics. We also describe novel connections to oriented matroids and positive geometry, as introduced by Arkani-Hamed, Bai and Lam.
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