  We show uniqueness of cylindrical blowups for mean curvature flow in all
dimension and all codimension. Cylindrical singularities are known to be the
most important; they are the most prevalent in any codimension. Mean curvature
flow in higher codimension is a nonlinear parabolic system where many of the
methods used for hypersurfaces do not apply and uniqueness of cylindrical
blowups remained a major open problem. Our results imply regularity of the
singular set for the system.
