7 problems

Let $\Lambda$ range over periodic configurations of arbitrary fixed density in phase space, and consider the associated quantum packing, covering, and paving problems. The hexagona…

Let $Q_2=Q_1*Q_1$, where $Q_1(x):=\chi_{[-1/2,1/2]}(x)$, and let $\mathcal{G}(Q_2,a,b)$ denote the associated Gabor system. Nielsen's conjecture. The Gabor system…

Lemvig and Nielsen's conjecture. The Gabor system $\mathcal{G}(Q_2,a,b)$ is not a frame along the hyperbolas

Universal Gaussian Gabor asymptotic conjecture. There exists a constant $C_\Lambda$, depending on the geometry of $\Lambda(1)$, such that

Strohmer's conjecture. As the critical density is approached from above, $\alpha\to1^+$,

Uniform erasure-robustness conjecture. Suppose that $|\Lambda|=O(M\log^\alpha M)$, where $\alpha\geq0$ is to be specified. Then, for every $p\in(0,1)$, $\Delta(p)\geq C$ with high…

Non-frame conjecture. The Gabor system $\mathcal{G}(B_2,a_0,b_0)$ is not a frame. Furthermore, it is not a frame along the hyperbolas