88 problems

Attractor invariants conjecture. The attractor invariants are given by

Let $(Q,\tau)$ be a general $\imath$quiver, allowing loops, and let the quantum symmetric pair and Hall algebra constructions associated with $(Q,\tau)$ be defined as in Theorem. H…

Let $A$ be the path algebra of an ad-quiver $(Q,\sigma)$ without oriented cycles. Let $\beta\in\Delta(Q)^+$ be a positive root, and let $p_F(M)$ and $p_\sigma(\beta)$ denote the pe…

Nekrasov's localized wheel-condition conjecture. The inclusion of the tripled-quiver CoHA image into the wheel-condition algebra becomes an equality after localization:

Nekrasov's wheel-condition generation conjecture. The localized wheel-condition algebra equals the localized spherical subalgebra:

Let $\{K_i\}_{i\in\mathbb{Z}_+}^{\tau}$ be a family of knots generated by $K_1$ by adding full twists, and let $Q_1:=Q_{K_1}$ be a quiver corresponding to $K_1$. Form an augmented…

Let $K$ be a knot with corresponding quiver $Q_K$. For each crossing of type $L_+$, let $K_{i,\tau}$, with $\tau\binom{}{}$ replaced by $tau eq$? be the knot or link obtained by ch…

Let $K$ be a knot and let $Q_K$ be its corresponding symmetric quiver. A full twist on any pair of strands of $K$ should correspond to an appropriate unlinking or linking operation…

Let $K$ be a knot and let $\overline{P}_r(a,q)$ denote its unreduced $S^r$-colored HOMFLY-PT polynomial. Let $Q_K$ be a symmetric quiver with $m$ nodes, adjacency matrix…

Let $Q$ be a finite quiver, let $\widehat{Q}$ denote its Kronecker square, and let $L_{\mathbb K}(Q)$ be the Leavitt path algebra over a field $\mathbb K$. Let…

Let $Q$ be a quiver, let $\widehat{Q}$ denote its Kronecker square, and let $L_{\mathbb K}(Q)$ be the Leavitt path algebra over a field $\mathbb K$. Let…

Let $H$ be a finite-dimensional weak bialgebra over an algebraically closed field $\mathbb K$ with commutative counital subalgebras. Let $Q$ be a finite quiver, and let…

Let $\Lambda$ be a finite dimensional algebra. Let $\mathsf{W}\subseteq\operatorname{mod}\Lambda$ be a $\tau$-perpendicular subcategory of $\operatorname{mod}\Lambda$, and let…

Let $(Q,\mathcal{P})$ be a thread quiver and let $\mathcal{C}$ be its path category. Write … for the category of pointwise finite-dimensional representations of $\mathcal{C}$. Here…

The ordered-quiver fruitfulness conjecture. The quivers $Q_n$ provide fruitful seeds, and any quiver obtained by gluing them according to an ideal triangulation of a bordered surfa…

The decorated-tiling fruitfulness conjecture. Any purely noncommutative decorated tiling of a polygon provides a fruitful seed. More generally, a decorated tiling of an arbitrary s…

Let $n$ and $p$ be as in the descent algebra setting, let mathop{\Lambda}\nolimits^+_p(n) denote the indexing set of $p$-regular partitions, and let $Q_{n,p}$ be the Ext quiver o…

Shuffle-algebra image conjecture.

Let $M$ be a knot complement, and let $\widehat{Z}(M)$ denote its homological block. Consider its positive or negative expansion, and let $\mathcal{P}_Q(\mathbf{x})$ denote a motiv…

Let $K$ be a knot, and let $P_K^{\mathcal{S}^r}(a,q)$ denote its colored HOMFLY polynomial in the totally symmetric representation $\mathcal{S}^r$. Define the HOMFLY generating fun…

Let $Q=(Q_0,Q_1)$ be a quiver, let $\lambda\in\mathbf{C}^{Q_0}$, and let $\Pi_Q^{\lambda}$ be the $\lambda$-deformed preprojective algebra. Write…

Weekly green path conjecture. There is a weekly green path from $v_0$ to $v$. In particular, the products

Brouwer–Haemers conjecture for non-magnetic quivers. For every $k\geq2$,

Let $G=\operatorname{SL}_n(\mathbb{C})$, let $v\leq w$ be elements of the Weyl group $W$, and let $Q_{\operatorname{Lec}}$ and $Q_{\operatorname{Ing}}$ be the quivers constructed f…

Let $Q$ be a quiver, let ${\mathcal B}_{\b0}$ be the slope-zero algebra graded by dimension vectors, and write … where $A_{Q,\boldsymbol{n}}(t)$ is the Kac polynomial. Define … and…