We prove the remaining absence-of-eigenvalues part of a conjecture of Avila and Jitomirskaya concerning the sharp spectral transition for completely resonant phases.
I will discuss some disordered quantum systems beyond the mean-field regime. In particular, delocalization for power-law random band matrices in the full regime of the decay exponent α > 0 of the variance profile (joint with Fan and Yang). The key challenge is to address the interplay between the non-mean-field nature of the model and the slow decay of the variance profile.