Hyperkahler Mirror Symmetry

The broad contours of mirror symmetry for Calabi-Yau manifolds are well-understood by now: classical complex-geometric information is exchanged for a symplectic-geometric expansion defined in terms of a mirror manifold. In the hyperkahler context, one may expect to make still stronger statements. In this talk, I will explain how to do so in the case of the K3 manifold (and some degenerations thereof). Namely, the exact Ricci-flat metric will have two expansions: (i) in terms of gauge theory on a four-torus, and (ii) as corrected by a list of enumerative invariants. In particular, these two constructions are the topics of the talks by M. Zimet and L. Fredrickson earlier in the workshop; I will review these constructions before explaining how they are related by this strong form of mirror symmetry. This talk surveys a series of joint works with L. Fredrickson, S. Kachru, A. Vasy, and M. Zimet.