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.  These Galois groups 
are difficult to determine, yet they contain subtle 
geometric information.

  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.) 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.