32 problems

Let $R$ be a Noetherian ring, complete with respect to a positive filtration topology, and let $(\sigma,\delta)$ be a continuous skew derivation compatible with the filtration. Def…

Let $k$ be the base field, let $q$ be the parameter, and let $\operatorname{\mathcal O}_q(M_{m,n}(k))$ denote the quantum coordinate ring of $m\times n$ matrices over $k$. A prime…

Goodearl–Lenagan's conjecture. There exists an explicit order-preserving bijection between

Let $R=k[x_1,x_2,\ldots,x_r]$ be a polynomial ring over a field. An ideal $I\subseteq R$ has the saturated chain property if, for every non-minimal associated prime ideal $Q$ of…

Let $G$ be a group satisfying the condition of Proposition, and let $B(G)$ denote the subgroup arising from the paper's $B(G)$-method; let $Z(G)$ be the centre of $G$. Centrality c…

Let $G$ be a $p$-valuable group, let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}$, and let $Z=Z(G)$ be the centre of $G$. For a prime ideal $P$ of…

Let $G$ be a $p$-valuable group, let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}$, and let $P$ be a prime ideal of the Iwasawa algebra…

Let $R$ be a ring and let $P$ be a height-one prime ideal in the setting under discussion. Principal height-one prime ideal conjecture. $P$ is principal. This conjecture is cited a…

Let $n\geq 1$, let ${\mathbb{B}}[x_1,\ldots,x_n]$ be the polynomial semiring over the Boolean semifield, and write $X^{\vec{a}}=x_1^{a_1}\cdots x_n^{a_n}$ for…

Alarcón–Anderson's classification conjecture. If $\mathcal{P}$ is a prime ideal in ${\mathbb{B}}[x]$ not containing $\langle x\rangle$, then…

Mascia–Rinaldo–Romeo conjecture. The following conditions are equivalent:

Let $n$ be a positive integer, let $\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$ be the space of symmetric $n\times n$ matrices, and for each symmetric matrix…

Let $k$ be an algebraically closed field, let $\mathcal{C}$ be a tensor category of moderate growth, and let $A\in\mathsf{Alg}\mathcal{C}$ be an algebra object. A prime ideal…

Let $W$ be the Witt algebra, let $\operatorname{U}(W)$ be its universal enveloping algebra, and let $\pi_\gamma$ be the family of representations defined in the paper, with kernels…

Let $W$ be the Witt algebra and let $\operatorname{U}(W)$ be its universal enveloping algebra. The Witt prime-to-primitive containment conjecture. Every proper prime ideal of…

The standardness conjecture. Let $G$ be a solvable, uniform pro-$p$ group, and let $P$ be a prime ideal in $\mathcal{O}G$. Then $P$ is virtually standard, and moreover if $p\in P$…

Three-line arrangement conjecture. In this example, the prime components correspond to point and line arrangements with at most three lines.

Let $R$ be a ring. A sequence $u_1,\ldots,u_n$ in $R$ is polynormal if $u_1$ is normal in $R$ and, for each $i=2,\ldots,n$, the image of $u_i$ is normal in…

Let $G$ be a split, simply connected, semisimple algebraic group over a field $K$ of characteristic zero, let $P_I$ be the standard parabolic subgroup associated with a set of simp…

Let ${\mathcal H}$ act on the quantum Grassmannian ${\cal O}_q(G_{mn}({\mathbb K}))$. For an ${\mathcal H}$-prime ideal, consider the quantum Plücker coordinates it contains; the P…

Let $G_{mn}^{ 0}( R)$ be the totally nonnegative Grassmannian, stratified by positroid cells, and let ${ H}\!\operatorname{-Spec}({\cal O}_q(G_{mn}({\mathbb K})))$ be the partially…

Catenarity conjecture. Every Noetherian unique factorization domain, and more generally every integrally closed domain, should be catenary.

Cauchon diagram covering conjecture. There exists a Cauchon diagram $w'\supsetneqq w$ such that

Normal separation conjecture. Normal separation holds for all quantum nilpotent algebras.

Let $K$) be a number field of bounded degree $n$ over $\mathbb{Q}$, with discriminant $D_K$ and class number $h_K$. An ideal class is said to contain a prime ideal if some prime id…