Joyce conjectures for Lagrangian mean curvature flow of surfaces

Lagrangian mean curvature flow is a potentially powerful tool for tackling problems related to symplectic topology.  Joyce made a serious of conjectures about the singularity formation, surgeries and long-time behaviour in Lagrangian mean curvature flow.  I will describe recent progress on these conjectures in the setting of 4-manifolds, where the Lagrangians are surfaces.