The Lagrangian mean curvature flow is the name given to the remarkable fact that mean curvature flow preserves the class of Lagrangian submanifolds in Kahler-Einstein manifolds. Upon hearing this fact, a natural question that springs to mind is whether there is a suitable boundary condition for this flow, such that the resulting flow with boundary still preserves the Lagrangian condition as in the closed case. Remarkably, standard Neumann and Dirichlet boundary conditions do not work, but there is a symplectically natural mixed Dirichlet-Neumann boundary condition involving a boundary Lagrangian flow which does!

In this talk I will introduce the subject of Lagrangian mean curvature flow, including the original proof of Smoczyk of preservation of the Lagrangian condition, and then explain recent results of myself, Lambert and Evans demonstrating the existence of a well-defined Lagrangian mean curvature flow with boundary.