The Optimal Littlewood-Richardson Homotopy

  Numerical homotopy methods for solving systems of 
polynomial equations may follow far more paths than
solutions, if the equations possess extra structure.
Devising methods to handle such structure has long
been a focus in the area, with the most well-known
being the polyhedral homotopies for sparse systems 
of polynomials.  This method is optimal for square
systems which achieve the BKK polyhedral bound.

   Algebraic geometry is a source of systems that 
have rich structure, and which may be overdetermined.
A particular well-studied class of such systems comes
from the Schubert calculus on the Grassmannian.
With Vakil and Verschelde, we proposed the optimal 
Littlewood-Richardson homotopy for Schubert calculus
which is based on Vakil's geometric Littlewood-Richardson
rule.  This is being implemented in Macaulay 2.
My talk will sketch this background, explain the
main features of this algorithm, and demonstrate
our current implementation.