I will close with a brief discussion of applications and future developments of the methods discussed here.

Recently a variant of the smoothing mechanism described here was used by myself, Hacking and Keel [26] to give a very general construction of mirrors of pairs $(Y,D)$ where $Y$ is a rational surface and $D$ is an effective anti-canonical divisor forming a cycle of rational curves. We make use of [27] to write down what we call the canonical scattering diagram, which can be described entirely in terms of the Gromov-Witten theory of the pair $(Y,D)$ (and more specifically, counts of curves intersecting $D$ at only one point). This scattering diagram determines the mirror family. However, there is an additional crucial tool used to partially compactify the family constructed. This is necessary because unlike the affine manifolds considered in this paper, the natural one to associate to the pair $(Y,D)$ has a singularity at a vertex of the polyhedral decomposition. There is no local model for a smoothing at this vertex, and as a consequence, one constructs families which are “missing” a point. To add this point back, one needs to be sure there are enough functions on the family constructed, and it turns out homological mirror symmetry suggests a natural way to construct such functions. This can be done tropically, creating what we call theta functions. The same construction applied to the case of degenerating abelian varieties indeed produces ordinary theta functions, and we anticipate the functions we construct in these other contexts will be similarly useful. See [34] for a survey of these ideas.

The construction of [26] then also solves a problem which pre-dates mirror symmetry. In particular, we prove a conjecture of Looijenga concerning smoothability of cusp singularities.

Theta functions can be viewed as canonical bases for rings of functions on an affine variety or spaces of sections of line bundles on projective varieties. As such, they make contact with canonical bases in cluster algebra theory, providing a framework for constructing canonical bases of cluster algebras.

We also expect that the techniques for surface pairs $(Y,D)$ will generalize. Indeed, a mirror partner to any maximally unipotent normal crossings degeneration of K3 surfaces can be constructed along similar lines, in work in progress with Hacking, Keel and Siebert. The expectation is that with an additional helping of log Gromov-Witten theory, one should be able write down a general construction in all dimensions for mirror partners to maximally unipotent degenerations of Calabi-Yau manifolds.

There still remains the question of extracting enumerative information from periods which provided the original excitement in mirror symmetry. Here we showed how enumerative geometry can be reflected in the mirror, but in a rather local way. We expect that it should be possible to carry out the computation of period integrals to extract genus zero Gromov-Witten invariants of the mirror, but some technical issues remain in this direction. Nevertheless, the program of understanding mirror symmetry via degenerations, inspired by the SYZ conjecture, seems to provide a powerful framework of thinking about mirror symmetry inside the realm of algebraic and tropical geometry.