On the Kotani-Last and Schrodinger conjectures
  In the theory of ergodic one-dimensional Schrodinger operators, ac spectrum has been traditionally expected to be very rigid. Two key conjectures in this direction state, on one hand, that ac spectrum demands almost periodicity of the potential, and, on the other hand, that the eigenfunctions are almost surely bounded in the essential suport of the ac spectrum. We show how the repeated slow deformation of periodic potentials can be used to break rigidity, and disprove both conjectures. 
Artur Avila
math.DS math-ph math.MP math.SP