Counting torus fibrations on a K3 surface
Among all complex two-dimensional manifolds, K3 surfaces are distinguished for having a wealth of extra structures. They admit dynamically interesting automorphisms, have Ricci-flat metrics (by Yau’s solution of the Calabi conjecture) and at the same time can be studied using algebraic geometry. Moreover, their moduli spaces are locally symmetric varieties and many questions about the geometry of K3s reduce to Lie-theoretic ones.
In this talk, I will discuss the analogue on K3 surfaces of the following asymptotic question in billiards - How many periodic billiard trajectories of length at most L are there in a given polygon ? The analogue of periodic trajectories will be special Lagrangian tori on a K3 surface. Just like for billiards, such tori come in families and give torus fibrations on the K3.