Degenerations of K3 Surfaces, Gravitational Instantons, and M-Theory

The detailed study of degenerations of K3 surfaces as complex manifolds
goes back more than forty years and is fairly complete. Much less is
known about the analogous problem in differential geometry of finding
Gromov--Hausdorff limits for sequences of Ricci-flat metrics on the K3
manifold. I will review work of H.-J. Hein and G. Chen--X. Chen on
gravitational instantons with curvature decay, and describe applications
to the K3 degeneration problem. I will also review some very recent work
of Hein-Sun-Viaclovsky-Zhang constructing some additional important
examples. The known examples are connected to physical dualities, as I
shall also explain.