8 problems

General support-pattern conjecture.

Kahle–Newman conjecture. For every finite abelian $p$-group $G$,

For each $n$, let $\Sigma_n=(\sigma_{n,1},\ldots,\sigma_{n,n})$, where each $\sigma_{n,j}$ is a subset of $[n]=\{1,2,\ldots,n\}$. Let $X_n\in\operatorname{M}_n(\mathbb{Z}_p)$ be Ha…

Cohen–Lenstra conjecture for $p=3$. $\operatorname{Conj}^{+}(3,\alpha)$ and $\operatorname{Conj}^{-}(3,\alpha)$ are true for every positive integer $\alpha$.

Let $X_\bullet$ be a random simplicial complex, and let $H_\ast(X_\bullet)_{\operatorname{tors}}$ denote the torsion subgroup of its homology. A Cohen–Lenstra distribution conjectu…

Let $\mathcal{T}^2(n)$ be the set of $2$-trees on $n$ vertices, and let $X$ be drawn uniformly from $\mathcal{T}^2(n)$. For a fixed prime $q$, let the Sylow $q$-subgroup of…

Let $LT(n)$ be the largest torsion group found in the first homology during the torsion burst of the Linial–Meshulam process. For a fixed prime $q$, let the Sylow $q$-subgroup of…

Let $G$ be a finitely generated pro-$p$ group. Let $\operatorname{Freq}(G)$ denote its limiting frequency among imaginary quadratic fields, and let $\operatorname{Meas}(G)$ denote…