Galois groups of Schubert problems

   Building on work of Jordan from 1870, in 1979 Harris 
showed that a geometric monodromy group associated to 
a problem in enumerative geometry is equal to the Galois 
group of an associated field extension.   Vakil gave a
geometric-combinatorial criterion that implies a Galois
group contains the alternating group.  With Brooks and 
Martin del Campo, we used Vakil's criterion to show that 
all Schubert problems involving lines have at least 
alternating Galois group.

  My talk will describe this background and sketch a 
current project to systematically determine Galois groups 
of all Schubert problems of moderate size on all small 
classical flag manifolds, investigating at least several 
million problems.  This will use supercomputers employing 
several overlapping methods, including combinatorial 
criteria, symbolic computation, and numerical homotopy 
continuation, and require the development of new 
algorithms and software.