We finish the proof of Shelah's 'Trichotomy' theorem.

I will review some classical and new results about finite dimensional integrable
Hamiltonian systems, emphasizing the interplay between symplectic geometry
and spectral theory. The talk is aimed at a general audience.

Abstract:
 We consider decaying oscillatory perturbations of periodic Schr\"odinger
 operators on the half line. More precisely, the perturbations we study
 satisfy a generalized bounded variation condition at infinity and an $L^p$
 decay condition. We show that the absolutely continuous spectrum is
 preserved, and give bounds on the Hausdorff dimension of the singular part
 of the resulting perturbed measure. Under additional assumptions, we
 instead show that the singular part embedded in the essential spectrum is
 contained in an explicit countable set. Finally, we demonstrate that this
 explicit countable set is optimal. That is, for every point in this set
 there is an open and dense class of periodic Schr\"odinger operators for
 which an appropriate perturbation will result in the spectrum having an
 embedded eigenvalue at that point.