Mean Curvature Flow (MCF) is a canonical way to deform sub-manifolds to minimal sub-manifolds. It also improves the geometric properties of sub-manifolds along the flow. The condition of being Lagrangian is preserved for smooth solutions of MCF in a Kahler-Einstein manifold. We call it Lagrangian mean curvature flow (LMCF) when requires slices of the flow to be Lagrangian.
Unfortunately, singularities may occur and cause obstructions to continue MCF in general. It is thus very important to understand the singularities, particularly isolated singularities of the flow. Isolated singularity models on soliton solutions that include self-similar solutions and translating solutions. In this talk, I will report some of my work with my collaborators on studying singularities of LMCF. It includes soliton solutions with different important properties and an in-progress joint project with Dominic Joyce that aims to understand how singularities form and construct examples to demonstrate these behaviours.