We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.
We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly @math 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 @math 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. Abstract Let G be a simple simply connected complex algebraic group. We give a Lie-theoretic construction of a conjectural mirror family associated to a general flag variety G P , and show that it recovers the Peterson variety presentation for the T -equivariant quantum cohomology rings q H T ∗ ( G P ) ( q ) with quantum parameters inverted. For SL n B we relate our construction to the mirror family defined by Givental and its T -equivariant analogue due to Joe and Kim.