Notes: Online seminar, also streamed in RSLT11.

Coassociatives are four-dimensional calibrated submanifolds of $G_2$ manifolds, seven dimensional manifolds with holonomy $G_2$. There is especially rich geometry to be studied when two coassociatives intersect transversely inside the ambient $G_2$ manifold. Non-compact examples of this phenomenon involving the Harvey-Lawson $Sp(1)$ invariant coassociatives in $R^7$ have been studied by Lotay and Kapouleas, who use the $U(1)$ symmetry in these examples to show the transverse intersection can be resolved by gluing in a family of Lawlor necks. The topology of the resulting coassociative is that of a generalized connected sum: the connected sum of the original two coassociatives along their intersection circles. In this talk, I will report on progress towards a generalization of this gluing construction for compact coassociative submanifolds intersecting transversely in an arbitrary $G_2$ manifold. If time permits, I will also describe an invariant for certain types of such transverse coassociative pairs.