7 problems

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…

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 $d\geq 4$, 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…

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…

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.