The Frobenius structure conjecture in dimension two

The Frobenius structure conjecture is a conjecture about the geometry of rational curves in log Calabi–Yau varieties proposed by Gross–Hacking–Keel. It was motivated by the study of mirror symmetry. It predicts that the enumeration of rational curves in a log Calabi–Yau variety gives rise naturally to a Frobenius algebra satisfying nice properties. In a joint work with S. Keel, we prove the conjecture in dimension two. Our method is based on the enumeration of non-archimedean holomorphic curves developed in my thesis. We construct the structure constants of the Frobenius algebra directly from counting non-archimedean holomorphic disks. If time permits, I will also talk about compactification and extension of the algebra.