288 problems

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…

Let $\Omega\subset\mathbb C^n$ be a smoothly bounded pseudoconvex domain with a non-compact automorphism group, and let $p\in\partial\Omega$ be a boundary orbit accumulation point,…

Let $A$ be a generic supersingular principally polarized abelian variety of dimension $g\geq 2$, and let $\mathbb Z_m$ denote the cyclic group of order $m$. Oort's conjecture. The…

Wilson's conjecture. These four sufficient conditions should also be necessary for a circulant graph to be nontrivially unstable.

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…

Hilbert-scheme genus conjecture. The variety $\Sigma(\mathcal{X})$ is a smooth curve of genus $3$ if and only if $\operatorname{Aut}(\mathcal{X})$ is finite.

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 … =^{-}oplusoplus^{+}. … , and let the diagram automorphism group of $^{+}$ have order $1$, $2$, or $3$. Andruskiewitsch–Dumas conjecture. The automorphism group of $U_q(^{+})$…

Mahalanobis's conjecture. Every Miller $p$-group is special for odd $p$.

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.

Let $\Bbbk$ be an algebraically closed field of characteristic zero and let $q\in\Bbbk^\times$ be a nonroot of unity. For $n\geq 2$, let $\mathcal O_q(M_n(\Bbbk))$ denote the singl…

Let $G$ be a finite graph whose Hadwiger number is at most $h$, and let $f$ be a function. A composition factor of a group is a factor in a composition series of that group. Babai'…

Non-regular-dessin conjecture. Every such uniform passport admits a non-regular dessin d'enfant with nontrivial automorphism group, that is, a dessin $\mathscr{D}$ satisfying

A group has an automorphism tower formed by repeatedly taking its automorphism group. Baumslag's general conjecture proposed that a sufficiently symmetric group should have a very…

Let $X$ be a rigid variety over an algebraically closed field of characteristic zero, meaning that $X$ admits no nontrivial actions of the additive group of the ground field. Write…

Let $q\in\mathbb{K}^\times$ be not a root of unity, and consider the quantized nilradical for $\mathfrak{g}=\mathfrak{sl}(n)$ and $J=\{\alpha_k\}$, which is isomorphic to the algeb…

Ghinelli's conjecture. The generalized quadrangle ${\mathcal S}$ does not admit a group of automorphisms acting regularly on its point set.

Gromov's conjecture. Geometric manifolds with large automorphism groups should be classifiable.

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…

Potočnik–Spiga–Verret's conjecture. A finite transitive permutation group is graph-restrictive if and only if it is semiprimitive.

Let $D$ be an irreducible bounded domain with a one-parameter group of holomorphic automorphisms and a compact quotient. Kazhdan's conjecture. The domain $D$ is biholomorphic to a…

The Johnson graph $J(n,i)$ has as its vertices the $i$-element subsets of $\{1,\ldots,n\}$, with two vertices adjacent when their intersection has cardinality $i-1$. Write…

Transposition conjecture.

Let $F_n$ be the free group of rank $n$, and let $E\Pi A(F_n)$ denote the group of elementary palindromic automorphisms of $F_n$. Collins's torsion-freeness conjecture. The group…

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