Berry & Robbins, in their discussion of the spin–statistics theorem in quantum mechanics, were led to ask the following question. Can one construct a continuous map from the configuration space of n distinct particles in 3–space to the flag manifold of the unitary group U(n)? I shall discuss this problem and various generalizations of it. In particular, there is a version in which U(n) is replaced by an arbitrary compact Lie group. It turns out that this can be treated using Nahm's equations, which are an integrable system of ordinary differential equations arising from the self–dual Yang-Mills equations. Our topological problem is therefore connected with physics in two quite different ways, once at its origin and once at its solution.