Quantization and tropical geometry for toric manifolds

Let X be a toric manifold. We will describe families of Kahler structures on X which are interesting from the point of view of geometric quantization. Along each family, the Kahler metric collapses in the Gromov-Hausdorff sense to a Hessian metric on the moment polytope P. Moreover, as the Kahler structure degenerates, the amoebas for hypersurfaces in X tropicalize under an appropriate Legendre transformation of P. This is based on joint work with T.Baier, C.Florentino and J.Mourao.