Toric degenerations of Grassmannians

Many toric degenerations and integrable systems of the Grassmannians Gr(2, n) are described by trees, or equivalently subdivisions of polygons. These degenerations can also be seen to arise from the cones of the tropicalisation of the Grassmannian. In this talk, I focus on particular combinatorial types of cones in tropical Grassmannians Gr(k,n) and prove a necessary condition for such an initial degeneration to be toric. I will present several combinatorial conjectures and computational challenges around this problem.  This is based on joint works with Kristin Shaw and with Oliver Clarke.