  We show that for each k and n, the cyclic shift map on the complex
Grassmannian Gr(k,n) has exactly $\binom{n}{k}$ fixed points. There is a unique
totally nonnegative fixed point, given by taking n equally spaced points on the
trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is
even). We introduce a parameter q, and show that the fixed points of a
q-deformation of the cyclic shift map are precisely the critical points of the
mirror-symmetric superpotential $\mathcal{F}_q$ on Gr(k,n). This follows from
results of Rietsch about the quantum cohomology ring of Gr(k,n). We survey many
other diverse contexts which feature moment curves and the cyclic shift map.
