Quasiperiodic Schrödinger operators are simple deterministic models of quantum motion that nevertheless exhibit strikingly complicated behavior: fractal spectra, metal–insulator transitions, and an unexpected sharp sensitivity to arithmetic properties of parameters. The central example is the almost Mathieu operator, whose study led to celebrated results such as the Ten Martini problem. But that model is highly special, and for a long time it was unclear whether these phenomena reflected genuine universality or exceptional symmetry.

I will discuss recent work showing that these spectral phenomena are robust and universal in a natural class. The key new insight is that one can uncover intrinsic low-dimensional symplectic dynamics hidden inside the problem, even when the standard dual cocycle formalism is no longer available. This leads to robust results on sharp arithmetic spectral transitions and the Ten Martini problem.

The talk will be aimed at a broad audience and will focus on the ideas and the emerging conceptual picture.