Galois groups of Schubert problems       

   In 1870 Jordan explained how Galois theory can be applied 
to problems from enumerative geometry, with the group encoding 
intrinsic structure of the problem.  Earlier Hermite showed 
the equivalence of Galois groups with geometric monodromy 
groups, and in 1979 Harris initiated the modern study of 
Galois groups of enumerative problems.  While difficult
to determine in general, several methods have been 
developed recently to partially determine Galois groups 
in the Schubert calculus.       

  I will describe this background and discuss a project
that seeks to determine Galois groups of all Schubert 
problems of moderate size, investigating millions of 
problems.   This project combines theoretical results
with supercomputers employing several overlapping methods, 
including combinatorics, symbolic computation, and numerical 
homotopy continuation.  It is driving the development 
of new algorithms and software, and a partial classification
is emerging from this study.