30 problems

Classification conjecture. The only possibilities for $\phi$ are

Analytical zero-divisor conjecture. If $\Gamma$ is torsion-free, then $\lambda_{\Gamma}(f)$ is injective for every non-zero $f\in\mathbf{C}\Gamma$. This is a classical conjecture c…

Hartley's conjecture. If the unit group $U(FG)$ satisfies a group identity, then $FG$ satisfies a polynomial identity.

Cameron's integral-domain conjecture. If $G$ has no finite orbits, then $A(G)$ is an integral domain.

Cameron's prime-element conjecture. If $G$ has no finite orbits, then $\varepsilon$ is a prime element in $A(G)$.

Let $G$ be a finite group, let $k$ be a field whose characteristic divides the order of $G$, and let $B$ be a block of the group algebra $kG$ with defect group $P$. Linckelmann's c…

Finite-Krull-dimension conjecture. The following assertions hold:

Semiperfectness criterion. The ring $\mathbb{Z}_{(p)}G$ is semiperfect if and only if

Commutativity conjecture. For fixed $n$, the elements $\mathbf{X}_{a,b}$ commute for all $a,b\in\mathbb{N}$; equivalently,

Let $\mathrm{G}$ be a finite group, let $p$ be a modular prime for $\mathrm{G}$, and let $\mathcal{L}_p[\mathrm{G}]$ be the associated modular Plesken Lie algebra. Let…

Let $G$ be a finite group, and let $\mathbb{C}[G]$ be its complex group algebra. A bialgebra is of Knutson type if every simple module over it has trivial Knutson index, meaning th…

Fix $\varepsilon>0$. Let $G$ be a finite group of order $n$, and say that $G$ is $n^\varepsilon$-quasirandom when every nontrivial irreducible complex representation of $G$ has dim…

Let $G$ be a finite group, let $M(G)$ denote its matching number, and define the minimum slice rank … where the minimum is over algebraically closed fields $k$ and $T_{k[G]}$ is th…

Integer-polynomial positivity conjecture. There is a polynomial $A{{boldsymbol{\lambda}}{boldsymbol{mu}}}^{{boldsymbol{nu}}}({bf q})inmathbb Z[{bf q}]$ such that

One-cycle product conjecture. (1) For every irreducible polynomial $f\in\mathbb F_q[t]$ of degree $d$ with constant term $-\xi a$, there exist $g,h\in G_d$ of modified types…

Stable-center generation conjecture. The stable center

Let $\mathbb{F}$ be a field and $G$ a torsion-free group. A unit of the group algebra $\mathbb{F}[G]$ is called trivial when it is a non-zero scalar multiple of a group element. Ka…

Let $\mathbb{T}_n$ be the group generated by the symbols $t_{ij}$ subject to the triangular relations, and let $\mathcal{A}_n$ be the associated algebra with unit group…

Let $E=(\mathbb{Z}/p)^2$ and let $kE$ be its group algebra. For a $kE$-module $M$ of constant Jordan type, let $\mathfrak{K}$ and $\mathfrak{I}$ denote the functors used to form th…

Let $G$ be a finite group, and let $k$ be a field whose characteristic divides the order of $G$. The ghost number finiteness conjecture. The ghost number of $kG$ is finite. The con…

Ghost-number extremality conjecture.

Let $F$ be a field and $G$ a countable group. The group algebra $F(G)$ is directly finite, meaning that for any $x,y\in F(G)$, $xy=1$ implies $yx=1$. Direct finiteness conjecture.…

Cyclotomic expansion conjecture. There exists a filtration-preserving algebra morphism

Let $G$ be a surface group, and let $\mathbb{C}[G]^h$ denote the hermitian elements of its group algebra. Write $\Sigma^2\mathbb{C}[G]$ for the cone of finite sums of hermitian squ…

Generalized Strong Bass conjecture. For each non-elliptic conjugacy class $\langle x\rangle$, the image of