  This paper introduces an intersection theory problem for maps into a smooth
manifold equipped with a stratification. We investigate the problem in the
special case when the target is the unitary group and the domain is a circle.
The first main result is an index theorem that equates a global intersection
index with a finite sum of locally defined intersection indices. The local
indices are integers arising from the geometry of the stratification.
  The result is used to study a well-known problem in chemical physics, namely,
the problem of enumerating the electronic excitations (excitons) of a molecule
equipped with scattering data. We provide a lower bound for this number. The
bound is shown to be sharp in a limiting case.
