16 problems

Polynomial bound conjecture. For each fixed $n$, the size of ${\rm SSS}(x)$ is bounded by a polynomial in the canonical length of $x$.

Let $P,Q\in\mathcal{O}$, let $h\in G$, and let $x\in\operatorname{GL}(4,q)$. Define … Regard the four equations in as simultaneous linear equations in the variables $\alpha^{nt}$ f…

Constructive membership testing difficulty conjecture. CMT is difficult whenever the group $G$ either involves a large abelian group as a quotient of a normal subgroup or has nonab…

Let $F_r$ be a free group of rank $r$, let $F_r^k$ denote the set of $k$-tuples of elements of $F_r$, and let $U_{\min}$ be a minimal representative of the orbit of $U$ under the r…

Sibling-pair and sibling-tuple conjecture. Among the $15$ groups of order $p^4$, there is exactly one sibling-pair. Among the $2p+2\gcd(p-1,3)+\gcd(p-1,4)+61$ groups of order $p^5$…

Let $n\geq 5$ and let $\beta\in B_n$ be a braid. A braid is almost strongly quasipositive (ASQP) if it is conjugate to an ASQP braid, and strongly quasipositive (SQP) if it is conj…

Let $G$ be a finite black-box group generated by $g_1,\ldots,g_k$, and let $h$ be a group element. The Group Non-Membership problem asks whether $h\notin G$. Aaronson–Kuperberg's c…

Let $d\geq 4$, let $q$ be a prime power, and let $\operatorname{SL}(d,q)$ be given by a generating set. Let $\xi$ denote an upper bound on the number of field operations for comput…

Let $k\geq 11$ and let $\mathcal{H}\subset \mathsf{UT}(n,\mathbb{Q})$ be any finite set of matrices. Write $\mathcal{I}=\{1,2,\ldots,k+1\}$, let $\operatorname{PI}^{\mathcal{I}}$ b…

Experimental stopping-index conjecture. The lower central series of ${\mathbf B}_{1,m}({\mathbb P}^2)$ always stops at

Let the two inputs be transitive tuples of permutations in the symmetric group, with tuple length $d$, and choose the permutations uniformly at random. Nearly-linear running-time c…

Let $\Gamma$ be the group under consideration, and let $\rho_k$ denote the representations indexed by $k$. An $\mathrm{SL}(n,p)$ quotient of $\Gamma$ is exposed by the family…

Eighth-power subgroup conjecture. If $G$ is any descendant of one of the $128$ candidates for $Q_3(G_K)$, then $G^8$ has index at most $2^{40}$ and is abelian.

Let $q$ be one of the field sizes considered for the Suzuki group, let $g$ be the element defining the cosets, and let $\langle g\rangle$ denote the subgroup generated by $g$. Cons…

The graph-theoretic construction produces graphs from principal systems of blocks and uses the graph-isomorphism algorithm to compute intersections of the corresponding wreath prod…

Let $\Gamma_2$ and $\Gamma_3$ be the pro-$2$ groups considered in the preceding descendant-tree analysis, and call a pro-$2$ group a descendant of one of these groups when it occur…