140 problems

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 $D$ 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…

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…

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.

Let $C$ be a generic supersingular hyperelliptic curve of genus $g$ over a field of characteristic $2$. Refined hyperelliptic Oort conjecture. If $g$ is not a power of $2$, then ……

Let $C$ be a generic hyperelliptic curve of genus $g$ and $2$-rank $0$ over a field of characteristic $2$. Generic automorphism conjecture for 2-rank-zero hyperelliptic curves. If…

Let $X$ be a Coble surface with Coble curve $C\simeq\mathbb{P}^1$, and let … be the restriction homomorphism. Coble's restriction-map problem. Is $\ker(\rho)$ trivial for a general…

Let the reduced finitary power monoid of the additive group $(\mathbb{Z},+)$ be the quotient of the finitary power monoid by translation equivalence, and let $X\mapsto -X$ denote t…

Let $\widehat{\mathfrak g}$ be the completed Kac–Moody algebra, with graded pieces $\mathfrak g_k$. Define … and let $\widehat{{U}}=\widehat{{U}}_1$ and $\mathcal H$ be as above. D…

Normalization conjecture. The group $\mathcal H$ normalizes $\widehat{{U}}$. This is one of the conjectures proposed for extensions of Scott's definition in the Kac–Moody setting;…

Tringali–Yan conjecture. If $S$ is a numerical monoid properly contained in $\mathbb{N}$, then the automorphism group of $\mathcal{P}_{0}(S)$ is the trivial group:

Let $A=\mathbb{C}\oplus\mathfrak{m}$ be a finite-dimensional local algebra of dimension $n$, where $\mathfrak{m}$ is its maximal ideal, and let $\operatorname{Aut}(A)^{\circ}$ deno…

Let $\bar m$ enumerate a tuple from a small model $M\preceq\mathfrak{C}$, and let $\mu\in\mathfrak{M}^{\operatorname{def}}_{\bar m}(\mathfrak{C},M)$ be fim over $M$. By the cited r…

Let $F_2$ be the free group on two generators. A group is fully residually finite simple if, for every finite subset of its nontrivial elements, there is a homomorphism to a finite…

The automorphism-group idempotent fim measure conjecture. The following conditions are equivalent: (i) $\mu$ is idempotent with respect to the new convolution product; (ii) $\mu$ i…

Let $\Sigma$ be a marked surface, let ${\mathcal A}_\Sigma$ be its noncommutative cluster algebra, and let $I_{p,1}(\Sigma)$ denote the set of the relevant once-punctured indices.…