We will review the fascinating history of the classical ten martini problem and then briefly introduce a new scheme towards the study of spectral theory of analytic quasiperiodic Schrodinger operators. The basic objects  include the connection between the dual Lyapunov exponents and Avila’s global theory, the partially hyperbolic structure of the dual  long-range operators and the intrinsic symplectic structure of the dual center which is robust under trigonometric polynomial approximations. We will show how these enter into the solution of the robust ten martini problem. This is based on several joint works with Svetlana Jitomirskaya, Jiangong You, and Qi Zhou.