30 problems

Let $Q$ be a quiver with no loops or $2$-cycles, and let $W$ be an infinitely mutable potential on $Q$. Write $\operatorname{\mathcal{H}}_{Q,W}$ for the cohomological Hall algebra…

Let $\widetilde Q$ be a quiver with potential $\widetilde W$, let $G$ be the relevant group, and let $\upzeta$ be a stability condition. Write $\mathtt{JH}$ for the semisimplificat…

Let $Q$ be a finite quiver, let $W$ be a polynomial potential on $Q$, and let $\operatorname{\mathsf{nil}}(\mathcal P(Q,W))$ be the category of finite-dimensional right modules ove…

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\mathbf{\Gamma}^{[a,b]}(\underline{w}_0)$ and…

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\widehat{\mathbf{\Gamma}}^{[a,b]}(\underline{w}_0)$ be the c…

Let $Q$ be a quiver and let $W$ range over quasi-homogeneous infinitely mutable potentials. Write $\operatorname{\mathcal{H}}_{Q,W}$ for the associated cohomological Hall algebra.…

Nilpotent independence conjecture. The nilpotent CoHA $\operatorname{\mathcal{H}}^{\operatorname{nilp}}_{Q,W}$ should be independent of $W$. This is proposed as a way to exclude th…

Let $Q$ be the quiver with potential considered in the paper, let $\operatorname{\mathcal{H}}_{Q,W}$ be its cohomological Hall algebra, and let $\mathcal{S}_Q$ be the spherical sub…

Let $\delta$ and $\check{\epsilon}$ be weights for a quiver with potential, and write $\check{f}_{\delta}$ and $f_{\check{\epsilon}}$ for the dual and ordinary tropical $F$-polynom…

Let $X$ be a toric Calabi–Yau 3-fold, and let $(Q,W)$ be a quiver with potential constructed from $X$ such that … Here, trivial attractor DT invariants means that all attractor Don…

A $3$-Calabi--Yau algebra is an algebra with the corresponding Calabi--Yau property, and a quiver with potential consists of a quiver together with a potential whose associated Gin…

Let $(Q,W)$ be a quiver with potential associated to a non-compact Calabi–Yau threefold, let $\Gamma$ be its charge lattice, and let $\langle-,-\rangle$ be the antisymmetrized Eule…

Let $X$ be a compact, acyclic, orientable circle bundle of dimension $n$, and let $k[pi_1(X)]$ denote its fundamental group algebra over a field $k$. A superpotential algebra is an…

Let $X$ be a compact, orientable $3$-dimensional manifold with contractible universal cover, and let $k[pi_1(X)]$ be its fundamental group algebra. A superpotential algebra is an a…

Generalization conjecture. These two theorems are special cases of more general results valid for such categories. The conjecture asks for extensions of both the map-to-Teichmüller…

Realizability conjecture. Each such category $\mathcal{C}$ can be realized as a generalized cluster category for a quiver with potential.

Let $Q$ be a quiver, and let $(\widetilde{Q},\widetilde{W})$ be the associated tripled quiver with potential. Let $\mathfrak{g}_Q$ be the Lie algebra associated to $Q$ in the Okoun…

Let $(Q,W)$ be a quiver with a generic potential, and suppose that its Jacobian algebra $J_{(Q,W)}$ is finite dimensional. Genteelness conjecture. The Jacobian algebra $J_{(Q,W)}$…

Let $(Q,W)$ be a quiver with superpotential of degree $N$, and let $\Gamma$ be its associated Ginzburg differential graded algebra. Write $\operatorname{CT}(Q,W)$ for the cluster b…

Let $(Q,W)$ be a quiver with potential, and consider the conclusions of the bimodule $3$-Calabi–Yau result [?]. In particular, these include the corresponding conclusions for the q…

Generic-stability Lie-superalgebra conjecture. If $\theta$ is generic, then $\mathcal{H}^{\theta\mhyphen\mathsf{ss}}_{Q,W,\mu}$ is the universal enveloping algebra of a Lie superal…

Fix a quiver $Q$, a potential $W$, a stability condition $\zeta$, and a slope $\mu$. Let $\Lambda_{\mu}^{\zeta}$ be the set of dimension vectors of slope $\mu$, let…

Let $\mathcal{A}$ be a category satisfying conditions (7) and (8), let $W$ be its potential, and let $\phi^{mot}$ denote the motivic vanishing-cycle realization. For each dimension…

Smith's conjecture. There is an equivalence of triangulated categories

Yangian conjecture. The critical CoHA of $(\widetilde{Q},W_Q)$ is isomorphic to the Yangian $\mathcal{Y}_Q$ associated with $Q$: