25 problems

Let $(A,G,\alpha)$ be a separable dynamical system, where $A$ is a $C^*$-algebra, $G$ is a locally compact group, and $\alpha$ is an action of $G$ on $A$. The system satisfies the…

Let ${\bf T}_n$ be the set of standard Young tableaux with $n$ boxes, let $I_T$ be the primitive ideal attached to $T$, let ${\mathcal V}_T$ be its orbital variety, and let…

Canonical left $\tilde{S}$-region conjecture. The ideal $J(\lambda)$ is completely prime.

Canonical cell conjecture. The element of minimal length is unique, and $J(\lambda)$ is a completely prime maximal ideal of ${\mathbf U}({^L \mathfrak{g}})$ whose associated variet…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\Ua(\g)$ be its enveloping algebra. ACC conjecture. The algebra $\Ua(\g)$ satisfies the ascending chain condition on primitive…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\Ua(\g)$ be its enveloping algebra. Complete-primality conjecture. Every primitive ideal of $\Ua(\g)$ is completely prime. The…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$. Let $\operatorname{P.Prim}\operatorname{S}(\g)$ denote the Poisson primitive spectrum of the symmetric algebra and let…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote the associated Poisson primitive ideals, and l…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be one-point local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote their associated Poisson primitive i…

Let ${\mathfrak g}_{\mathbb C}$ be a reductive Lie algebra over ${\mathbb C}$ with Chevalley basis ${\mathcal B}$, let $R={\mathbb Z}[1/q\mid q$ is a bad prime for $\Phi]$, an…

Let $G_{\mathbb R}$ be a real reductive group, let $\widetilde{\mathcal O}$ be a birationally rigid nilpotent cover for its complexification, and let $\mathcal O_{\theta}$ be a $K$…

Let $K$ be a field, let $\mathcal{G}$ be an ample groupoid, and let $u\in\mathcal{G}^{(0)}$. Write $A_K(\mathcal{G})$ for the Steinberg algebra, $\mathcal{G}_u$ for the isotropy gr…

Vogan's quantization conjecture. For each such cover $\widetilde{\mathcal{O}}$, there is a canonically defined completely prime primitive ideal $I_0(\widetilde{\mathcal{O}})$ in th…

Let $\mathfrak g$ be one of $V\!ir$, $W$, or $W_{\geq -1}$. For a local function $\chi\in\mathfrak g^*$, choose a polarization and form the annihilator of the induced module; write…

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…

Let $W$ be the Witt algebra and $W_{\geq -1}$ its subalgebra spanned by the basis elements of index at least $-1$. Let $\operatorname{U}(W)$ and $\operatorname{U}(W_{\geq -1})$ be…

Let $\mathfrak{g}$ be a simple Lie algebra, let $U(\mathfrak{g})$ be its universal enveloping algebra, and let $\mathscr{I}_0$ be the set of primitive ideals with trivial central c…

Let $\mathcal{O}$ be a nilpotent orbit, let $S_{\mathcal{O}}$ be its special piece, and let $\overline{A}(\mathcal{O})$ be its Lusztig quotient. For…

Let $\mathbb{O}$ be a special nilpotent orbit, let $\lambda$ be a dominant representative of a regular central character, and let $W_\lambda$ be its integral Weyl group. Let $W_I$…

Let $G$ be a complex semisimple algebraic group, and let $\mathcal{O}_q(G)$ be the corresponding quantized function algebra. Let $\mathbb{C}[G]$ carry the standard Poisson structur…

Premet's conjecture. The map

Premet's uniqueness conjecture. For any primitive $\mathcal J\in\mathfrak{Id}_{\mathbb O}(\mathcal U)$, the set of primitive ideals…

Primitive-proportion conjecture. As $n\rightarrow\infty$, the proportion of $\mathcal{H}$-primes that are primitive tends to

Exponential enumeration conjecture. For every positive integer $n$,

Primitive-prime enumeration conjecture. The number of primitive $\mathcal{H}$-primes in ${\mathcal O}_q(M_{3,n})$ is