Singularities of Lagrangian Mean Curvature Flow

In a Calabi-Yau manifold, mean curvature flow--the downward gradient for the area functional--preserves the Lagrangian condition. Thus Lagrangian mean curvature flow suggests a way to find minimal Lagrangian submanifolds of a CY manifold, provided the flow lasts for all time. However, finite-time singularities are expected along the flow; in fact, ill-behaved singularities are generic in some sense. In this talk we will discuss two main results: one, that type I (mild) finite-time singularities can be predicted by looking the cohomology of the initial Lagrangian submanifold, and two, that type II (ill-behaved) singularities can be modeled as unions of special Lagrangian cones. We will also discuss what these results say about using mean curvature flow to understand the topology of Lagrangian submanifolds.