14 problems

Let $G$ and $G'$ be the symmetry groups specified in the claim, and write $M_G(a,b,c)_{-1}$ and $A_G(a,b,c)_{-1}$ for the corresponding matrices specialized at $q=-1$. Let…

Let $M$ be a nonsingular integer matrix, let $P$ be the relevant finite set of perfect matchings, and let … with the twisted commutation relation determined by the pairing…

Let $G$ be a planar graph for which the paper's spanning-tree construction produces a planar graph $G'$, and let the tree group of $G$ denote the cokernel associated with its spann…

Mészáros's critical-distribution conjecture. For every finite abelian $p$-group $G$,

Let $p$ be a prime, let $(\nu_{\alpha})_{\alpha\in\mathbb{R}}$ be a proposed one-parameter family of distributions on the set of finite abelian $p$-groups, and let $n_1<n_2<\dots$…

For every $n\geq 1$, let $\bX$ be a $\{0,1\}^{n\times n}$-valued random matrix with independent entries, and let $b=e$ be the all-ones vector. Let $\bW$ denote the associated walk…

Let $P(t) \in \mathbb{Z}_p[t]$ be a non-constant monic square-free polynomial, written as … where each $P_j(t)\in\mathbb{Z}_p[t]$ is monic and its reduction $\bar{P}_j(t)$ modulo…

Let $X_m:=\{x_1,\ldots,x_m\}$ be a finite ordered subset of $\mathbb{Z}_p$ whose elements have distinct reductions modulo $p$, and let…

Cyclic cokernel conjecture. Under the uniformity assumption that, for each prime $p$, the entries of $M_n$ are uniformly distributed modulo $p$, and the independence assumption tha…

Let $M_1,\ldots,M_k$ be independent random matrices in the setting of Theorem, and consider the chain of quotient maps … For Haar matrices, and universally in the setting of Theore…

Fix finitely many monic polynomials $P_{1}(t),\dots,P_{l}(t)\in\mathbb{Z}_{p}[t]$ whose images in $\mathbb{F}_{p}[t]$ modulo $p$ are distinct and irreducible. For $1\leq j\leq l$,…

Asymptotic independence conjecture. We have

Let $P_{1}(t),\dots,P_{l}(t)\in\mathbb{Z}_{p}[t]$ be monic polynomials whose images modulo $p$ are distinct and irreducible in $\mathbb{F}_{p}[t]$. Let $q_{j}=p^{\deg(P_{j})}$, let…

Let $P_{1}(t),dots,P_{l}(t)\in\mathbb{Z}_{p}[t]$ be monic polynomials whose images modulo $p$ are distinct and irreducible in $\mathbb{F}_{p}[t]$. For each $j$, let $G_{j}$ be a fi…