Joint work with Paul Hacking (U Mass Amherst). We will outline a proof of homological mirror symmetry for K3 surfaces, in the following setting: on the symplectic side, we have any K3 surface (X, \omega) with \omega integral Kaehler; and on the algebraic side, we have any K3 surface Y with Picard rank 19. Our approach includes establishing a conjecture of Lekili-Ueda for `type III' degenerations of K3 surfaces, which may be of independent interest. Time allowing, we will explain what generalisations we expect. The talk will be targeted at an audience with a broad range of geometric interests.
