Aspects of Grassmannian scattering amplitudes

This talk will consist of two parts. In the first part I will review how to compute n-point scattering amplitudes in the $\phi^3$ theory from three different approaches. The first approach is using Feynman diagrams, which is the standard way to compute these observables in physics. Then I will explain how the tropical Grassmannian, TropG(2,n), is closely related to the space of the Feynman diagrams for this theory, and I will propose a formula as an integral over TropG(2,n) to compute first $\phi^3$ and then $\phi^p$ scattering amplitudes, for any p>2. The third approach is the CHY formula, which is an integral over the moduli space of n points on $CP^1$. In the second part of the talk, motivated by the isomorphism between the Grassmannians G(2,n) and G(n-2,n), I will explain how to generalize the CHY formula to an integral over the space of n points on higher dimensional projective spaces $CP^{k-1}$, thus providing a natural generalization of the notion of scattering amplitudes. I will also point out connections with higher dimensional tropical Grassmannians and will comment on some of the features of these new objects.