Mirror Symmetry for P and Tropical Geometry

In [13, 14, 15, 16], Bernd Siebert and myself have been working on a program designed to understand mirror symmetry via an algebro-geometric analogue to the Strominger-YauZaslow program [29]. The basic idea is that the controlling objects in mirror symmetry are integral affine manifolds with singularities. One can view an integral affine manifold as producing a mirror pair of manifolds, one a symplectic manifold and one a complex manifold, each of twice the real dimension. These correspond to the Aand B-models of mirror symmetry. A great deal of the work carried out by myself and Siebert has been devoted to building up a dictionary between geometric notions on affine manifolds and objects in the Aand B-models. If mirror symmetry is to become self-evident from this process, one should be able to find a single geometric notion on an affine manifold which corresponds to both rational curves on the A-model side and corrections to period calculations on the B-model side. A conceptual proof of mirror symmetry would identify these objects in the world of integral affine geometry. (For a survey of this basic approach, the reader may consult [12]; however, while this paper is motivated by this program, it is largely self-contained.)