Galois groups for 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.  These subtle geometric 
invariants are difficult to determine.  A consequence of 
Vakil's geometric Littlewood-Richardson rule is a 
combinatorial criterion that the Galois groups of a Schubert 
problem on a Grassmannian contains at least the alternating 
group, and Vakil showed that most problems on small 
Grassmannians satisfy his criterion.

  Exploiting Harris's equivalence, Leykin and I used numerical 
homotopy continuation to compute Galois groups of problems 
involving mostly divisor Schubert classes, finding all to 
be the full symmetric group.  (This included one problem 
with 17589 solutions.)

  My talk will describe this background and sketch a current 
project with Leykin, undergraduate students, graduate students, 
and postdocs 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.