  This extended Oberwolfach report (to appear in the proceedings of the MFO
Workshop 2335: Aspects of Aperiodic Order) announces the full solution to the
Dry Ten Martini Problem for Sturmian Hamiltonians. Specifically, we show that
all spectral gaps of Sturmian Hamiltonians (as predicted by the gap labeling
theorem) are open for all nonzero couplings and all irrational rotations. We
present here the proof strategy.
