  We first bound the codimension of an ancient mean curvature flow by the
entropy. As a consequence, all blowups lie in a Euclidean subspace whose
dimension is bounded by the entropy and dimension of the evolving submanifolds.
This drastically reduces the complexity of the system. Combined with
\cite{CM12}, this gives the first general bounds on generic singularities of
surfaces in arbitrary codimension.
  We also show sharp bounds for codimension in arguably some of the most
important situations of ancient flows. Namely, we prove that in any dimension
and codimension any ancient flow that is cylindrical at $-\infty$ must be a
flow of hypersurfaces in a Euclidean subspace. This extends well-known
classification results to higher codimension.
  The bound on the codimension in terms of the entropy is a special case of
sharp bounds for spectral counting functions for shrinkers and, more generally,
ancient flows. Shrinkers are solutions that evolve by scaling and are the
singularity models for the flow.
  Finally, we show rigidity of cylinders as shrinkers in all dimension and all
codimension in a very strong sense: Any shrinker, even in a large dimensional
space, that is sufficiently close to a cylinder on a large enough, but compact,
set is itself a cylinder. This is an important tool in the theory and is key
for regularity; cf. \cite{CM8}.
