Mean curvature flow deforms a submanifold in the direction of its mean
curvature vector. When the initial submanifold is Lagrangian in a
Kahler-Einstein manifold, the solution will also be Lagrangian whenever
it is smooth. It thus becomes a nice way to construct special Lagrangians.
However, finite-time singularities may occur in general and cause the main
difficulties. I will report some of my works on special solutions to
Lagrangian mean curvature flow that is closely related to the study of singularities. A
big part of the talk will concentrate on examples related to Schoen-Wolfson
cones, which are the obstructions to the existence of special Lagranians in
two-dimension.