Camille Jordan observed that Galois groups arise in enumerative geometry, and we now also understand them as monodromy groups. Recent advances in theory and technology have enabled the study of these Galois groups in the context of the Schubert calculus for Grassmannians. A picture is emerging from this study of an apparent dichotomy; all known Schubert Galois groups are either the full symmetric group or are imprimitive, and hints of a classification are emerging. Recently, Esterov considered this question for systems of sparse polynomials and proved this dichotomy in that setting. While this classification identifies polynomial systems with imprimitive Galois groups, it does not identify the groups.

I will sketch the background, including some of the results in the Schubert calculus before explaining Esterov's classification and ongoing work identifying some of the imprimitive Galois groups for polynomial systems.