8 problems

Let $(\Gamma,L)$ be an $n$-dimensional periodic graph, where $\Gamma$ is a directed graph, $L$ is a free abelian group of rank $n$ acting freely on $\Gamma$, and the quotient graph…

Korotyaev–Saburova's conjecture. The perturbed operator $\mathcal{A}_\Gamma+Q$ should have no flat bands.

No-flat-band conjecture. The maximal abelian cover of $G$ should have no flat bands.

Let $\Gamma$ be a $\mathbb{Z}^d$-periodic graph with one crossing edge per generator, let $H$ be a periodic self-adjoint operator acting on $\ell^2(\Gamma)$, and let…

The periodic quantum graph is the truncated hexagonal tiling, and the preceding construction gives eigenfunctions associated with the factors $S(a,\rho)=0$, $S'(a,\rho)=-2/3$, and…

Let $G$ be a general $\mathbb{Z}^2$-periodic graph, and consider the spanning-forest measure with slope parameter. A configuration has zero slope when its slope is $0$. Zero-slope…

Let $G$ be a general $\mathbb{Z}^2$-periodic graph, and consider recurrent random walks on directed graphs killed at a wired boundary. Let the Green's function describe the corresp…

Let $T$ be a tile, and let $c_n(T)=\operatorname{cr}(\bigcirc(T^n))$. The limit … exists. Rationality conjecture for crossing limits. There exists a computable function $f$ that as…