Explore a lecture on the SYZ-conjecture in mirror symmetry and its relationship to Calabi-Yau hypersurfaces in toric Fano manifolds. Delve into Yang Li's work that reduces a weak version of the SYZ-conjecture to solving a real Monge-Ampère equation on polytope boundaries. Learn about the varying solvability of this equation across different families and understand its connection to optimal transport theory. Examine how optimal transport theory provides insights into the discriminant locus of the PDE. Based on collaborative research with Rolf Andreasson, Mattias Jonsson, Enrica Mazzon, and Nick McCleerey, discover the current challenges and open problems in this mathematical domain.