18 problems

Non-termination conjecture. The LCS of ${\mathbf{B}}_{2,m}({\mathbb{P}^2})$ does not stop.

Let $\Gamma_n$ be the pure cactus group, let $\Gamma_n^k$ denote its lower central series, and let $\mathcal{L}_n$ be the associated graded Lie algebra after quotienting by $2$-tor…

MLS LCS formula. If ${\mathcal A}$ is an MLS arrangement with group $G$, then, for all $k\ge 2$,

Let $\mathcal{A}$ be a complex hyperplane arrangement with arrangement group $G=G(\mathcal{A})$. Let $R_1(\mathcal{A})=\bigcup_i L_i$ be its resonance variety, and set…

Let $F_n$ be the free group of rank $n$, let $\mathrm{IA}_n=\mathcal{A}_n'(1)$ be the group of automorphisms of $F_n$ acting trivially on its abelianization, and let … be the lower…

Let $A_n$ be the free associative algebra on $n$ generators, and let $j=(2,\dots,2)\in\mathbb{N}^k$. All-two tuple conjecture. … The formula is motivated by combinatorial considera…

Weil-generic vanishing conjecture. For Weil generic $P$ of degree $d$,

Even-index product conjecture. If $i,j\geq 2$ are both even, then in general

Let $A=\mathbb{Z}\langle x,y\rangle/(x^m+y^m)$, and let $N_3(A)$ denote the third lower central series quotient, with $N_3(A)[d]$ its degree-$d$ component. Rank formula conjecture.…

Let $A_n(\mathbb Z)$ be the free associative algebra on $n$ generators, and let $B_\ell(A_n(\mathbb Z))[m_1,\ldots,m_n]$ denote the multihomogeneous component of multidegree…

Let $A_n(\mathbb F_p)$ be the free associative algebra on $n$ generators over the finite field $\mathbb F_p$, and let $B_2(A_n(\mathbb F_p))[m_1,\ldots,m_n]$ denote its multihomoge…

Composition-factor conjecture. The vectors $v_{k,0}$ and $z_{k,0}$ are non-zero in $B_3(A'_{2n})$, and the $H_{2n}$-module composition factors of $B_3(A'_{2n})$ are

Euler-operator injection conjecture. The operator $D$ is an injection

Let $\Gamma=\Gamma_{\mathcal T}$ be one of the groups introduced in Section 4 of the cited source, associated to a prime power $q=p^3$ with $p\neq 3$. Let $\gamma_i(\Gamma)$ denote…

Let $G$ be the group considered in the paper, let $H_i$ be the associated upper-triangular subgroups, and let $\lambda_i(G)$ and $\gamma_i(G)$ denote the terms of the lower exponen…

Let $A$ be an associative algebra, and let $L_1(A)=A$ and $L_{k+1}(A)=[A,L_k(A)]$ denote its lower central series. A pair $(i,j)$ is called null when, for every associative algebra…

Geometric characterization conjecture. A knot $K$ is $n$-trivial for all $n\in\mathbb{N}$ if and only if it is $n$-hyperbolic for all $n\in\mathbb{N}$.

Let $p$ be an odd prime, and let $\Gamma_p$ be the corresponding generalised Fabrykowski-Gupta group. The width-two conjecture. Then $\Gamma_p$ is a group of width $2$. The conject…