24 problems

Let $X$ and $Y$ be colored graphs. Write $X*Y$ for the colored graph obtained by putting a copy of $X$ at each vertex of $Y$, let $\mu(Z)$ denote the spectral measure of the charac…

Randomised twisted Toeplitz spectral-measure conjecture. The measures $\mu_n$ converge weakly to

Banded perturbation conjecture. The measures $\mu_n$ converge weakly to

Let $(\mu,\Lambda)$ be a singular continuous spectral pair in $\mathbb{R}$. A real number $t\in\mathbb{R}$ is called a spectral eigenvalue of $(\mu,\Lambda)$ if $(\mu,t\Lambda)$ is…

Closed-curve non-spectrality conjecture. The measure $mu$ is not spectral.

Support conjecture for tight-frame spectra. Then

Łaba–Wang conjecture. If $\mu_{\rho,\mathcal{D},{\bf p}}$ is a spectral measure, then ${\bf p}$ is equal-weight, so $p_j=1/m$; $\rho=1/N$ for some integer $N$; and there exists…

Exponential localization and symmetry conjecture. The densities of states $\nu^{\gamma,\beta}_{\mathrm{INB},+,k}$ and $\nu^{\gamma,\beta}_{\mathrm{INB},\times,k}$ exist and satisfy…

Let $A$ satisfy the conditions of Theorem 3.1, and let $A'$ be the matrix constructed in Lemma 3.4. For the Toeplitz coefficients $a_j$, define the limiting measure $\mu$ as the di…

Let $\boldsymbol{\kappa}_\lambda$ be the nonnegative left eigenvector of the dual multiple stochastic matrices associated with a measure $\mu$ and parameter $\lambda$. A mass point…

Let $M_N$ be a sequence of unitarily invariant random matrices, let $\lambda=(\lambda_1,\ldots,\lambda_N)$ be the eigenvalue vector of $M_N$, and let $\mu=\pi_{N,N-1}\lambda$ be th…

Conjecture on singular A-L measures. There exists a Dirichlet-regular Widom set ${\mathsf{E}}$ with DCT such that A-L holds and $\mu^\alpha$ has a nontrivial singular component wit…

Let $f$ satisfy Assumptions and, and suppose there exist $\alpha\in(0,2)$ and $r_0>0$ such that … for all $|x|<r_0$. Let $D\subset\mathbb{R}^2$ be an open rectangle centred at the…

Let $\mu(N_n,B_n)$ be the Cantor–Moran measure associated with a sequence $\{(N_n,B_n,L_n)\}$ forming a Hadamard triple tower, and let $\nu_{>n}$ denote the pulled-back tail measur…

Bauer–Golinelli conjecture. The following phase transition occurs: if $c<e$, then $\mu_c$ has no extended states at $0$; if $c>e$, then $\mu_c$ has extended states at $0$.

Bauer–Golinelli conjecture. For any $c\in(0,\infty)$,

Fix an integer $d\ge 3$, and let $A^{(d)}_n$ be uniformly drawn from the set $\mathcal{A}_{n,d}$ of adjacency matrices of $d$-regular directed graphs on $n$ vertices. Let…

Let $\mu\in\mathcal M_1(\mathbb T)$ have Verblunsky coefficients $(\alpha_k)_{k\geq 0}\in\mathbb D^{\mathbb N}$. Let $\theta_{\tt g}$, $N^-$, $N^+$, $\lambda_i^+$, $\lambda_i^-$, a…

Let $\mathcal{P}$ be the collection of probability measures on the unit ball $B(1)\subseteq\mathbb{R}^{2}$, and let $c_{\max}$ denote the maximal value attained by the Nazarov–Sodi…

Let $\mathcal{P}_{Symm}$ be the family of symmetric probability measures on $\mathcal{S}^{1}$, and let $d_{\max}$ denote the maximal value attained by the Nazarov–Sodin constant…

Let $\Gamma$ be a finite subgroup of $GL(2,\mathbb{Z})$, let $(\mathcal{G}^{\Gamma}_{\rho_1},\mathcal{G}^{\Gamma}_{\rho_2})$ be the associated pair of graphs, let $\mathbb{T}^2$ be…

Let $T$ be a rank one transformation with a fixed sequence of tower partitions $(\xi_n)$ such that $\xi_{n+1}$ refines $\xi_n$ and every measurable set $A$ can be approximated by a…

Let $M$ be the lognormal multifractal random measure with parameter $\gamma^2$, and let the extended singular integral be defined, for $x\in[0,1]$ and continuous $f$, by … Let…

The integer-operator conjecture for group-measure-space rings. Every normal operator in this subring is an integer operator. This is the group-measure-space analogue of the conject…