K3 surfaces: dynamics and moduli spaces

A compact complex surface that has a nowhere vanishing holomorphic 2-form, and which is simply-connected, is called a K3 surface. The geometry and dynamics of K3s is rich: they admit Ricci-flat metrics and have homogeneous parameter spaces, analogous to Teichmuller and moduli spaces of Riemann surfaces. Additionally, K3s often admit interesting automorphisms about which many questions remain open. I will first provide the necessary background, following an analogy with Riemann surface equipped with a holomorphic 1-form. Then I will discuss some results regarding the interaction of automorphisms and Ricci-flat metrics on K3s. Joint work with Valentino Tosatti.