We discover a striking duality, T-duality, between two seemingly unrelated objects. The hypersimplex $\Delta_{k+1,n}$ is a polytope obtained as the image of the positive Grassmannian $\mbox{Gr}^{\geq 0}_{k+1,n}$ under the well-known moment map. Meanwhile, the amplituhedron $\mathcal{A}_{n,k,2}$ is the projection from the positive Grassmannian $\mbox{Gr}^{\geq 0}_{k,n}$ into the Grassmannian $\mbox{Gr}_{k,k+2}$ under the amplituhedron map. Introduced in the context of the physics of scattering amplitudes, it is not a polytope and has full dimension inside $\mbox{Gr}_{k,k+2}$. We draw novel connections between the two objects and prove many new properties of them. We exploit T-duality to relate their triangulations and generalised triangles -- maximal cells in a triangulation. We subdivide $\mathcal{A}_{n,k,2}$ into chambers as $\Delta_{k+1,n}$ can be subdivided into simplices -- both enumerated by Eulerian numbers. Moreover, we prove a main result about the hypersimplex and the positive tropical Grassmannian $\mbox{Trop}^+ \mbox{Gr}_{k+1, n}$, several conjectures on the amplituhedron, and find novel cluster-algebraic structures.

This talk is based on the recent joint work with L. K. Williams and M. Sherman-Bennett (Preprint, arXiv: 2104.08254).