Entropy and codimension bounds for generic singularities

We show that all closed $2$-dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subspace of small dimension. The entropy and dimension of the subspace are both $\leq C\,(1+\gamma)$ for some universal constant $C$ and genus $\gamma$. These are the first general bounds on generic singularities in arbitrary codimension.