Mirror symmetry for minuscule flag varieties

Abstract: We prove cases of Rietsch mirror conjecture that the A-model of projective homogeneous varieties is isomorphic to the B-model of its mirror, which is a partially compactified Landau--Ginzburg model constructed from Lie theory and geometric crystals. The conjecture relates to deep objects in algebraic combinatorics. Our method of proof comes from Langlands reciprocity, and consists in identifying the quantum connection as Galois and the crystal as automorphic. I will mention further potential interactions between symplectic geometry and number theory that it suggests. Work with Thomas Lam.