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.

For a wide class of 2D periodic elliptic operators, we show that the minima of band functions can only be attained on a discrete set of values of quasimomenta. The talk is based on joint results with Nikolay Filonov.

Localization is well established in the standard Anderson model in the strong
disorder phase. On the other hand, the motivation for the problem, which lies in
many body systems, still lacks a developed theory. We will discuss progress in this
direction, in particular the state labeling method recently developed by Imbrie [I].
As is typical in proof of localization by multiscale analysis, an apriori estimate
to control spectral properties was required, in this case a limited level attraction
estimate. The estimate remains unproven and appeared in [I] as a physically
reasonable assumption. A key difficulty in some many body models such as quantum
spin models is the non monotonicity of spectral energies with respect to random
parameters.
We address this issue in the simplest possible setting, we consider a single body
model with bare energies depending analytically on the random parameters. In
multichannel Schrodinger models, the potentials at each site of the lattice are
matrices which may depend analytically on the random parameters. We will discuss a
method for controlling level attraction which allows a multiscale localization
proof which does not utilize resolvent methods. Our main result is a limited level
attraction estimate [IM] similar to that which appears in [I] as an assumption.

This talk is based on joint work with John Imbrie.

[IM] Imbrie, John Z., and Rajinder Mavi. "Level Spacing for Non-Monotone Anderson
Models." arXiv preprint arXiv:1506.06692 (2015).

[I] Imbrie, John Z. "On many-body localization for quantum spin chains." arXiv
preprint arXiv:1403.7837 (2014).