As is well known, a complex m × m matrix A is a commutator (i.e., there are matrices B and C of the same dimensions as A such that A = [B, C] = BC − CB) if and only if A has zero trace. If ∥ · ∥ is the operator norm from ℓ_2^m to itself and | · | any ideal norm on m × m matrices then clearly for any A, B, C as above |A| ≤ 2∥B∥|C|. Does the converse hold? That is, if A has zero trace are there m × m matrices B and C such that A = [B, C] and ∥B∥|C| ≤ K|A| for some absolute constant K? If not, what is the behavior of the best K as a function of m? I’ll talk mostly on a couple of years old result of Johnson, Ozawa and myself which gives some partial answers to this problem for the most interesting case of | · | = ∥ · ∥. The solution is closely related in both directions to the best possible estimates in the Kadison–Singer problem. Time permitting I’ll also speak on a more recent result for | · | = the Hilbert–Schmidt norm.