140 problems

Let $\Gamma$ be a random labeled graph on $n$ vertices, let $\operatorname{Aut}^{\pi}(\Gamma)$ denote its two-fold automorphism group, and let $\mathbb{E}$ denote expectation over…

Automorphism-group conjecture.

Let $G$) be a connected graph with $n\geq 5$ vertices and prime factor decomposition … where $r>1$. Let $\psi$ be the homomorphism from $\operatorname{Aut}(G)$ to…

Automorphism-group conjecture. For $q>r>\frac{n(n-1)}{2}$, the automorphism group of $SRM_q[n,r]$ contains a subgroup isomorphic to $M$; equivalently, $SRM_q[n,r]$ is invariant und…

Let $\widetilde{\mathscr{D}}$ be a design and let ${\mathscr{D}}(\widetilde{\mathscr{D}},(e_1,e_2))$ denote the design obtained from it by the rectangular combination construction,…

Let $\widetilde{\mathscr{D}}$ be the input design, let $e$ and $\widetilde{k}$ be the parameters of Construction $\mathscr{D}(\widetilde{\mathscr{D}},e)$, and let…

Infinite motion conjecture. If $X$ is a connected, locally finite graph with infinite motion, then

Let $X$ be an affine algebraic variety defined over an algebraically closed field, and let $G$ be a subgroup of $\operatorname{Aut}(X)$ generated by a finite collection of…

Let $S$ be a smooth, affine, cubic surface whose automorphism group is infinite and discrete. A classification conjecture asserts that $S$ is either a Markov-type surface, or the c…

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…

Babai's motion conjecture. There exists an $\alpha>0$ such that either

Let $\Omega$ be a bounded pseudoconvex domain with $C^\infty$ boundary, and let $\mathcal{L}(\Omega)$ denote the limit set of its automorphism group. A point…

Yan's conjecture. If $d$ is a simple derivation of $R_n$, then $\operatorname{Aut}(R_n)_d$ is conjugate in $\operatorname{Aut}(R_n)$ to a subgroup of the group of translations.

Grid-graph equitable-partition conjecture. The equitable partitions of $P_m\oblong P_n$ are the orbit partitions of natural actions of the subgroups of the automorphism group of th…

Let $D$ be a derivation as in case $c)$ of Theorem, and let ${\rm Aut}(D)$ denote its automorphism group. Finiteness conjecture. In case $c)$ of Theorem, the group ${\rm Aut}(D)$ i…

Loewy filtration destabilisation conjecture. If $(X,L)$ has non-reductive automorphism group, then the Loewy filtration destabilises $(X,L)$.

Let $\mathbb{K}$ be a field, let $q\in\mathbb{K}^*$ be not a root of unity, and let ${\mathcal O}_q(M_n)$ be the quantisation of the ring of regular functions on $n\times n$ matric…

Boundedness of $B$-canonical representations. There exist positive integers $B_l$ and $B'_l$ such that

Properness conjecture. The action of $\scr A(M)$ on $M$ should be proper unless $M$ is CR equivalent to the sphere or to the Heisenberg group with its standard CR structure.

Small-index conjecture. If $H$ is a subgroup of $Autf(M)$ with index strictly less than $2^{\aleph_{0}}$, then there exists a finite subset $A\subset M$ such that

Let $C$ be a one-point algebraic-geometric code associated to the curve … over $GF(p^2)$, with length $n=2p(p-1)+p$, and let $X$ be the corresponding curve over a field $F$. Arrang…

Let $X$ be a projective Calabi–Yau manifold, meaning that its canonical bundle is torsion and … and let $\mathcal U$ be a universal torsor for $X$. Lift conjecture. The action of…

Work over an algebraically closed field $\mathbb K$ of characteristic zero. Let $Y$ be an affine variety, and let $\mathbb{G}_{\rm a}$ and $\mathbb{G}_{\rm m}$ denote the additive…

Restriction conjecture. Every automorphism of $\mathcal P(H)$ maps $\mathcal P_1(H)$ bijectively onto itself and therefore restricts to an automorphism of $\mathcal P_1(H)$.

Inner automorphism conjecture. The automorphisms of $\mathcal P(\mathbb Z)$ are all inner, and the automorphisms of $\mathcal P_0(\mathbb Z)$ are all inner.