In this talk I will report on joint work with Rosa Sena-Dias studying minimal Lagrangian submanifolds appearing as the fibers of the moment map on a toric Kahler manifold. I will use action-angle coordinates to answer the following questions: How many such minimal Lagrangian tori exist? Can their stability, as critical points of the area functional, be inferred from the ambient geometry? Which sets of such Lagrangian submanifolds can be made minimal with respect to some toric Kahler metric.
If time permits I will also make some comments on Lagrangian mean curvature flow and Hamiltonian stationary Lagrangians.
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