Title: Twisted connected sum constructions of $G_2$ manifolds

At present, there are only two known constructions for $G_2$ manifolds: desingularizations of toroidal orbifolds (constructed by Joyce), and the twisted connected sum construction (originally introduced by Kovalev, and extended by Corti, Haskins, Nordström and Pacini). The latter construction begins from algebraic geometry -- a semi-Fano 3-fold and a K3 surface on it -- and also involves a nontrivial application of the "matching" of two non-isomorphic algebraic K3 surfaces whose underlying Ricci-flat metrics are the same.

Although we will be applying results from the theory of K3 surfaces which were covered in earlier lectures this fall, this lecture is independent of the earlier ones.