I will discuss a proposed method for answering a basic question of mirror symmetry. As currently understood, affine manifolds underly mirror symmetry between Calabi-Yau manifolds. From an affine manifold with certain properties, one should be able to construct a complex manifold and a symplectic manifold which are mirror to each other. Geometric structures on the affine manifold then translate into geometric structures on both the complex and the symplectic manifold, providing an explanation for much that is observed in mirror symmetry.

However, currently we do not know how to pass from an affine manifold to a complex manifold in key cases. I will discuss a method of doing this which is almost complete and in addition explain some of the correspondence between geometric structures mentioned above.