Generalized Witness Sets

I will describe how some ideas from intersection 
theory are useful in numerical algebraic geometry.  
The fundamental data structure in numerical algebraic 
geometry is that of a witness set, which is considered 
to be an instantiation of Weil's notion of a generic 
point.  Reinterpreting a witness set in terms of 
duality of the intersection pairing in intersection 
theory leads to a generalization of the notion that 
makes sense on many spaces and leads to a general 
notion of a witness set.  I will also describe how 
rational equivalence is linked to homotopy methods.

These notions are most productive for homogenous 
spaces, such as projective spaces, Grassmannians, 
and their products.  After explaining the general 
theory, I will sketch what this means for the Grassmannian.  
This is joint work with Bates, Hauenstein, and Leykin.