8. Conclusions

Building on Grassmannian formulations for scattering amplitudes in planar \(\mathcal {N}=4\) SYM – introduced by Arkani-Hamed et al. and Bullimore et al.—and on Hodges’ idea that amplitudes are ‘volumes’ of some geometric object, Arkani-Hamed and Trnka arrived at the definition of the amplituhedron in 2013. The above geometrisation programme—formalized in the framework of positive geometries—hinges on the idea that quantum observables in particle physics and cosmology come from underlying (novel) mathematical objects. Physical properties (e.g. locality and unitarity) purely emerge from combinatorics and geometry. Understanding this process advances our grasp of the basic principles of Quantum Field Theory and allows us to perform calculations which were previously beyond reach. Crucially, it also cross-fertilises ideas in pure mathematics, such as in algebriac combinatorics. Our work concerns understanding the combinatorics of tilings, T-duality and the cluster structures of ‘amplituhedra’—the positive geometries relevant for scattering amplitudes of \(\mathcal {N}=4\) SYM (and beyond). In this chapter we review the motivation and the results of our work, and present future directions.