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 Leykin, Martin del Campo, Vakil and Verschelde, 
we described and implemented the Littlewood-Richardson 
homotopy for Schubert calculus, which is based on 
Vakil's geometric Littlewood-Richardson rule.  
This has two independent implementations in Macaulay 2
and in PHCPack.   My talk will sketch this background
and explain the main features of this algorithm.