Galois groups in Enumerative Geometry and Applications
 
    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.  He posited that 
 a Galois group should be `as large as possible' in that it
 will be the largest group preserving internal symmetry in 
 the geometric problem.

   I will describe this background and discuss some work
 in a long-term project to compute, study, and use Galois
 groups of geometric problems, including those that arise    
 in applications of algebraic geometry.  A main focus is  
 to understand Galois groups in the Schubert calculus, a 
 well-understood class of geometric problems that has long
 served as a laboratory for testing new ideas in enumerative 
 geometry.