Degenerations of Kahler forms on K3 surfaces, and some dynamics

K3 surfaces have a rich geometry and admit interesting holomorphic automorphisms. As examples of Calabi-Yau manifolds, they admit Ricci-flat Kähler metrics, and a lot of attention has been devoted to how these metrics degenerate as the Kähler class approaches natural boundaries. I will discuss how to use the full automorphism group to analyze the degenerations and obtain certain canonical objects (closed positive currents) on the boundary. While most of the previous work was devoted to degenerating the metric along an elliptic fibration (motivated by the SYZ picture of mirror symmetry) I will discuss how to analyze all the other points. Time permitting, I will also describe the construction of canonical heights on K3 surfaces (in the sense of number theory), generalizing constructions due to Silverman and Tate. Joint work with Valentino Tosatti.