4 problems

Dry ten Martini conjecture. The spectrum of the almost Mathieu operator is a Cantor set and all allowed gaps are present.

Let $\Omega$ be the tiling space, let $\Xi$ be its canonical transversal, and let $\mu$ be an invariant probability measure on $\Omega$ inducing a transverse measure $\mu^t$ on…

Let $\Lambda$ be the quasicrystal with transversal $\Omega_{trans}$, and let $\mathcal{A}_{\theta=0}$ be its twisted groupoid $C^*$-algebra for zero magnetic field. The trace induc…

Global gap-labeling conjecture. The weak-coupling restriction, namely the existence of $\delta>0$ requiring $0<\left\Vert(\mathfrak{p},\mathfrak{q})-(1,0)\right\Vert<\delta$, can b…