Tropical Lagrangian sections and Looijenga pairs

We will discuss the first steps in an approach to proving homological mirror symmetry for Looijenga pairs through tropical Lagrangian sections. Namely, we will see how to construct these Lagrangian sections from tropical data corresponding to line bundles on the mirror and include them in a version on the Fukaya-Seidel category. Moreover, the Lagrangian Floer coholomogy of certain sections corresponds with integral points of polytopes that encode theta functions on the mirror. This is joint work with Abigail Ward.