|Location:||MSRI: Simons Auditorium|
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 purely algebro-geometrically. 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 some analogies between K3 and flat surfaces. The analogue of closed trajectories in flat geometry are special Lagrangian tori inside a K3 surface. Just like in the flat world, these come in families and give a fibration of the K3. Such fibrations arise in integrable systems and mirror symmetry. I will explain an asymptotic count for the number of fibrations, considered in appropriate families.