Abstract: "Batyrev and Borisov used the duality of Gorenstein cones to construct mirror pairs of Calabi-Yau varieties. Gross and Katzarkov and I make sense of a duality of arbitrary polyhedral cones as mirror symmetry and in particular provide mirror duals all hypersurfaces in projective toric varieties. This construction was used by Seidel and many others. I will explain in detail how it works for curves. Perverse sheaves of vanishing cycles play a key role."