Mirror symmetry & Looijenga's conjecture

A cusp singularity is an isolated surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In the 1980's Looijenga conjectured that a cusp singularity is smoothable if and only if the minimal resolution of the dual cusp is the anticanonical divisor of some rational surface. This conjecture can be related to the existence of certain integral affine-linear structures on a sphere. Existence of such integral-affine structures follows from constructions originally discovered in symplectic geometry.