In this talk we show that if the normalized cocycle related to a quasiperiodic Jacobi operator H_{v,c} (where v being the potential and c being the off-diagonal term) is reducible to a constant rotation for almost all energies with respect to the density of states measure, then its dual model has either purely point spectrum for almost all phase or purely absolutely continuous spectrum for almost all phase, depending on the winding number of c. As a corollary, we obtain the complete phase-transition of extended Harper's model in the positive Lyapunov exponent region.

Many-body localization (MBL) generalizes Anderson localization to interacting many-body systems. MBL challenges many fundamental notions of statistical physics. In this talk I will introduce what MBL is from a math-phys perspective. Then I will discuss some striking difference between Anderson localization and MBL, in particular the existence/absence of a mobility edge, i.e. a critical energy (or energy density for many-bod physics) that separates localized states from extended states. (Joint work with W. De Roeck, M. Müller, M. Schiulaz.)

Abstract:  For the second  phase transition line $\lambda=e^{\beta}$ of  almost Mathieu operator, we prove that for dense $\alpha$, the operator has purely singular continuous spectrum for every phases, and  for dense $\alpha$, the operator has pure point spectrum for almost every phases. This is joint work with Artur Avila and Svetlana Jitomirskaya. 

We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension at least 3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n>=3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. This is joint work with Yaiza Canzani, Rod Gover and Raphael Ponge. If time permits, we shall discuss related results for operators on graphs.