47 problems

Let $\mathcal{L}_N$ be Kac's master-equation operator on $L^2\left(\mathbb{S}^{N-1}\left(\sqrt{N}\right),d\sigma_N\right)$, and let … be its spectral gap. Kac's uniform spectral-ga…

Let $G=\mathrm{PSL}_2(\mathbb{R})^d$ and let $\Gamma$ be a lattice in $G$. For a bounded family $\mathfrak{B}\subseteq\widehat{G}$ of non-tempered irreducible unitary representatio…

Let $(G_n)_{n\geq 1}$ be a sequence of finite rooted graphs, and let $t^n_{\textup{mix}}(\varepsilon)$ be the total-variation mixing time of the continuous-time simple random walk…

Ferromagnetic ordering conjecture. The spectral gap $\gamma_k$ is independent of $k$; equivalently,

Super approximation conjecture. The group $\Gamma$ has the super approximation property with respect to all positive integers if and only if $\mathbb G_0$ is perfect.

Let $\mathcal{G}$ be a finite universal gateset, and say that $\mathcal{G}$ has a spectral gap when the spectral-gap property used in the paper's random-circuit equidistribution ar…

Let $G=G_1\times\dots\times G_l$ be a direct product of simple groups, let $\Gamma$ be an irreducible lattice in $G$, and let $\rho_0$ be the regular representation of $G$ on…

Let $F_n$ be the free group of rank $n$, let $K$ be the compact group appearing in the paper, and let $\operatorname{Hom}(F_n,K)/K$ denote the quotient of the representation variet…

Let $n\geq 2$, let $\phi_1,\ldots,\phi_n$ be a collection of elements of $SU(2)$, and let $L^2_0(SU(2))$ denote the orthogonal complement of the constant functions in $L^2(SU(2))$.…

Let $G$ be any graph equipped with a ferromagnetic Ising model, with couplings $J_{xy}$, and let $\tau_2$ denote the relaxation time of its Glauber dynamics. Peres's monotonicity c…

Let $\Lambda$ be a finite graph, and let $L$ be the generator of the symmetric simple exclusion process with positive jump rates $J_{xy}$ on its edges. Let $H_\Lambda^{(n)}$ be the…

Affleck–Kennedy–Lieb–Tasaki conjecture. These models should have a unique ground state in the thermodynamic limit and a spectral gap.

Let $n\geq 3$ and let $\vec{p}$ be a regular parameter vector. A label is neutral if $p_{c,i}=p_{i,c}=1/2$ for every $i\neq c$. Let … be the number of neutral labels, and let…

Fill's characterization conjecture. The following are equivalent:

Fill's spectral gap conjecture. Among all regular parameter vectors, the spectral gap is minimized by the uniform vector:

Mohammadi–Oh's conjecture. The optimal condition $\delta_\Gamma>d/2$ suffices for $L^2(\Gamma\backslash G)$ to admit a strong spectral gap for the complementary series, in arbitrar…

Step-size scaling conjecture. The $O(h^4)$ scaling of the spectral gap is merely an artifact of the isoperimetric approach.

Let $\Gamma$ be an irreducible higher-rank lattice. Lubotzky's property $\tau$ conjecture. The group $\Gamma$ has property $\tau$. This is listed among consequences of the spectral…

Let $\Gamma$ be a center-free irreducible higher-rank lattice in $G$, and let $V$ be a unitary representation of $\Gamma$ whose action does not extend to a $G$-action on any non-tr…

Let $\Gamma$ be a center-free irreducible higher-rank lattice in $G$, and let $V$ be a unitary representation of $\Gamma$. Assume that the $\Gamma$-action on $V$ does not extend to…

Let $M_\gamma$ be the periodic TASEP Markov matrix, and let $\lambda_2(\gamma)$ denote its second-largest eigenvalue. For sufficiently small $|\gamma|$, the largest eigenvalue is o…

Let $\Omega$ be a finite symmetric subset of $\operatorname{GL}_n(\mathbb{Z}[1/q_0])$, let $\Gamma=\langle\Omega\rangle$, and let $\pi_m$ denote reduction modulo $m$. Consider the…

Erbar–Fathi conjecture. The concentration assumption implies

Qualitative optimality conjecture. For at least some $d70b_0$ in the class $d70b_{d70b_0,k}$, the result cannot be qualitatively improved.

Almost-Diophantine conjecture. Every non-degenerate probability measure is almost Diophantine for some positive constants $c_1,c_2$.