The almost Mathieu cocycle with $\lambda>1$ represents a prototypical hyperbolic analytic cocycle, and at $\lambda=1$ a prototypical critical one in the sense of Avila's global theory. Hyperbolic Schrodinger cocycles, at energies in the spectrum are non-uniformly hyperbolic (in fact, spectral measures are supported on points where, at a given phase, the Oseledets multiplicative ergodic theorem does not hold coherently in both directions). The critical case represents the boundary between hyperbolicity and reducibility.

It is also a model heavily studied in physics literature and, besides one Fields medal, it is linked to several Nobel prizes. We will describe several recent results on this model, that resolve some further long-standing conjectures pertaining to both its non-uniformly hyperbolic and critical cases.

The talk is partially based on papers joint with Wencai Liu and Igor Krasovsky.