Galois groups in enumerative geometry

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.