23 problems

Let $q$ be the number of Potts symbols, let $\theta=\tanh(\beta)$, and let $d_0$ denote the random offspring degree under the tree measure $Q$. Reconstruction is the problem of rec…

Let $G=\operatorname{Sym}_n$ and let $T$ be the set of transpositions, so that $\operatorname{Sym}_n(T)$ is the corresponding Cayley graph, $B_r(\sigma)$ denotes the radius-$r$ bal…

Let $A$ and $B$ be two hypomorphic matrices, meaning that $B=\Sigma(A)$ for a hypomorphism $\Sigma$ as in the surrounding reconstruction setting. Let $\lambda_i$ be a simple eigenv…

Let $S_n$ be the permutation group on $n$ letters and let $\mathbb Z_2^n$ denote the group of diagonal sign matrices. Twisted-product refinement. The group $G(A)$ in the spectral r…

Let $A$ be a real $n \times n$ symmetric matrix. For each $i$, write $A_i$ for the matrix obtained by deleting the $i$th row and column, and let $\operatorname{eigen}(A)$ denote th…

Let $M$ be a channel and $T_b$ the $b$-ary tree. For $n$ and $m$, let $\sigma_{n,m}$ be obtained from the level-$n$ configuration $\sigma_n$ by applying $M^m$ independently to each…

Potts monotonicity conjecture. If reconstruction is solvable for $M_1$, then it is also solvable for $M_2$.

Consider the symmetric Potts channel $M$ on $q\geq 3$ symbols, with $b$-ary tree branching factor $b$ and second eigenvalue $\lambda_2(M)$. Potts-model below-bound conjecture. Ther…

Let $M$ be a channel with second eigenvalue $\lambda_2(M)$, and let $T_b$ be the $b$-ary tree. The reconstruction problem is census-solvable when the root can be reconstructed from…

Let $V\subseteq\mathbb{R}$ with $|V|=n$, and let the known distances be distributed as in $G(V,\lambda/n)$ for a parameter $\lambda>0$. Call a subset reconstructible if every injec…

Let $n\geq 4$, and let $G_1,\dots,G_k$ be a collection of $n$-edged graphs for some $k\geq 1$. Write $\coprod_{i=1}^k\operatorname{ED}(G_i)$ for the disjoint union of their edge de…

Let $k\geq 1$ and let $T\in\operatorname{YT}(n)$, where $n\geq k+4$. Here $\operatorname{mM}_k(T)$ denotes the multiset of $k$-minors of the standard Young tableau $T$. Multiset re…

Let $k\geq 2$ and let $T\in\operatorname{YT}(n)$, where $n\geq k^2+2k$. Here $\operatorname{M}_k(T)$ denotes the collection of $k$-minors of the standard Young tableau $T$. Reconst…

Sharp threshold conjecture. The threshold for the existence of a reconstructible set of size linear in $n$ is sharp and occurs at

Let $J(n,q)$ be the random $(n,q)$-jigsaw, and let reconstructibility mean that the jigsaw is uniquely determined by its deck. The critical number of colours is conjectured to be a…

Let $V(\phi)$ be a shift-invariant space, let $k$ and $\nu$ be positive integers, and let $\\{x_i:i\in\mathbb{Z}\\}$ be a separated set. For a function $f$, consider the nonuniform…

Let $G(n)$ denote the largest integer $k$ such that every partition of $n$ is uniquely determined by its multiset of $k$-minors, with the precise definition of $G(n)$ as in the pap…

Let $\mathcal{M}$ be a relational structure and let $G=\operatorname{Aut}(\mathcal{M})$. Let $F_n(G)$ denote the relevant orbit-counting sequence, and suppose that $F_n(G)$ is know…

Let $T$ be an infinite tree with root $\rho$. Assign the root a uniformly random label in $\{\pm1\}$ and propagate labels along edges, retaining the parent label with probability…

Let $T$ be an infinite tree with root $\rho$. In the two-state broadcast process, each vertex label $\tau_v\in\{\pm 1\}$ is generated by assigning the root a uniformly random label…

Let row sums be $\mathcal{R}=(r_1,r_2,\ldots,r_m)$ and column sums be $\mathcal{C}=(c_1,c_2,\ldots,c_n)$, where … and … Assume that the line sums are consistent. Small-boundary rec…

Mézard–Montanari's conjecture. If $q\leq 4$ and $d<d_{\max}$, then there is reconstruction if and only if

Equality conjecture. We believe that