The planarity estimate in higher codimension mean curvature flow

In this talk, we will discuss the mean curvature flow of $n$-dimensional submanifolds in $\mathbb{R}^N$ satisfying a pinching condition $|A|^2 < c |H|^2$ introduced by Andrews and Baker ('10). We will introduce a planarity estimate that shows singularities of these flows must become codimension one then survey more recent results and remaining open problems in this area.