17 problems

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…

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:

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…

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 $N\in\mathbb Z^+$ be the number of ball-piston pairs aligned along a one-dimensional chain, and consider the associated piston model. Spectral-gap conjecture. The spectral gap…

Boundary Gibbs-state conjecture. The parent Hamiltonian of the PEPS is gapped if and only if the boundary states can be written as Gibbs states of one-dimensional short-range Hamil…

Let $G$ be a compact simple Lie group, let $(g_1,\ldots,g_k)\in G^k$ be a random $k$-tuple, and let $\Gamma(g_1,\ldots,g_k)<G$ be the subgroup it generates. Gamburd–Jakobson–Sarnak…

Let $n$ be a positive integer with $n\ge 3$. For each $A\subset\{1,\ldots,n\}$, let $S_{n,A}$ be the set of permutations in $S_n$ fixing every element outside $A$, and define … For…

Let $X$ be a centered log-concave random vector in $\mathbb R^n$, let $\lambda_X^2$ be the largest eigenvalue of its covariance matrix, and let $g:\mathbb R^n\to\mathbb R$ be local…

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

Let $k\geq 2$, let $\mathcal S_k$ be the symmetric group on $k$ letters, and let $g:\mathcal S_k\to\mathbb R$ be any function. For nonnegative numbers…