We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.Michel van Garrel KIAS,discretizing Hitchin's Legendre duality.Enumerative aspects of the Gross-Siebert program 3 Kontsevich and Soibelman [32] demonstrated how one could reconstruct a K3 surface from an affine structure with singularities on S 2 . Using logarithmic geometry, Gross and Siebert were able to solve the reconstruction problem [20] in any dimension, obtaining a degenerating family of Calabi-Yau manifolds X → D over a holomorphic disk from the information of (B, P, ϕ) and a log structure. Furthermore, this family is parametrized by a canonical coordinate (in the usual sense in mirror symmetry). The construction features wall-crossings and scatterings, structures that encode enumerative information linking symplectic with complex geometry via tropical geometry. As will be hinted at in this exposition, Gromov-Witten theory [21] can also be incorporated in this framework. Toric conventionsWe assume familiarity with toric geometry. The interested reader is referred to the excellent exposition of Fulton [10]. As the following story is closely tied to toric geometry, it is convenient to begin by making a few conventions regarding notation.SetFor n ∈ N, set n, m to be the evaluation of n on m. Set a toric fan Σ in M R . Let Σ [n] signify the set of n dimensional cones of Σ . Let X Σ be the toric variety defined by Σ .Denote by T Σ the free abelian group generated by Σ [1] . For ρ ∈ Σ [1] , denote by v ρ the corresponding generator in T Σ . We will need the mapwhereρ is the integral vector generating ρ, that is ρ ∩ M = Z ≥0ρ .

We modify Gross's construction of mirror symmetry for P 2 [10] by introducing a descendent tropical Landau-Ginzburg potential. The period integrals of this potential compute a modification of Givental's J-function, explicitly encoding a larger sector of the big phase space. As a byproduct of this construction, new tropical methods for computing certain descendent Gromov-Witten invariants are defined.

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We use a formula of Bultot to compute the motivic zeta function for the toric degeneration of the quartic K3 and its Gross-Siebert mirror dual degeneration. We check for this explicit example that the identification of the logarithm of the monodromy and the mirror dual Lefschetz operator works at an integral level.