Most of the research in my group addresses fundamental issues concerning quantum dynamics in the presence of disorder, and topics related to critical phenomena.
In recent years one main focus was the influence of interactions in disordered systems. In a first step, the proof of the persistence of a localised phase at large disorder or weak interaction was achieved in systems of finitely, but arbitrarily many interacting particles.
More recently, we studied the phenomenon of resonant delocalisation, that is, the formation of extended states through resonating local quasi-modes in systems with an exponential growth of the volume. So far, we have rigorously established the existence of such modes in toy models as the Anderson model on the Bethe lattice or the complete graph. Ultimately it a challenge to see whether this sheds some light on ongoing debates concerning the behaviour of eigenfunctions such as non-thermalization for more realistic many particle systems.
Another current project concerns a proof of the ground-state phase transition in the quantum random energy model.