xiv INTRODUCTION B \ B0 relatively small (e.g., codimension at least two). How do we deal with this? By 2000, it was certainly clear to many of the researchers in the field that holomorphic curves in X(B) should correspond to certain sorts of piecewise linear graphs in B. Kontsevich suggested the possibility that one might be able to actually carry out a curve count by counting these graphs. In 2002, Mikhalkin [79, 80] announced that this was indeed possible, introducing and proving curve-counting formulas for toric surfaces. This was the first evidence that one could really compute invariants using these piecewise linear graphs. For historical reasons which will be explained in Chapter 1, Mikhalkin called these piecewise linear graphs “tropical curves,” introducing the word “tropical” into the field. This brings us to the following picture. Mirror symmetry involves a relationship between two different types of geometry, usually called the A-model and the B- model. The A-model involves symplectic geometry, which is the natural category in which to discuss such things as Gromov-Witten invariants, while the B-model involves complex geometry, where one can discuss such things as period integrals. This leads us to the following conceptual framework for mirror symmetry: A-model Tropical geometry B-model Here, we wish to explain mirror symmetry by identifying what we shall refer to as tropical structures in B which can be interpreted as geometric structures in the A- and B-models. However, the interpretations in the A- and B-models should be different, i.e., mirror, so that the fact that these structures are given by the same tropical structures then gives a conceptual explanation for mirror symme- try. For the most well-known aspect of mirror symmetry, namely the enumeration of rational curves, the hope should be that tropical curves on B correspond to (pseudo)-holomorphic curves in the A-model and corrections to period calculations in the B-model. The main idea of my program with Siebert is to try to understand how to go between the tropical world and the A- and B-models by passing through another world, the world of log geometry. One can view log geometry as half-way between tropical geometry and classical geometry:

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