13 problems

Let $Q$ be a quiver, let $d$ be an indivisible dimension vector such that … where $(d,e_i)$ is the relevant bilinear form and $Q_0$ is the vertex set. Let $\mathcal M_d^\chi(Q)$ be…

Let $Q$ be the quiver, let $d$ be the dimension vector, and let $\delta$ be as in Theorem; write $\operatorname{supp}\delta$ for its support, let $d_i$ denote the components of $d$…

Let $Q$ be a quiver, let $\mathbf{r}\in\mathbb{Z}_{\geq0}^{Q_{0}}$ be a rank vector, and let $\alpha\geq1$. Let $A_{(Q,\alpha),\mathbf{r}}$ be the associated Kac polynomial for the…

Let $Q$ be an arbitrary finite quiver and fix a dimension vector $\mathbf{d}\in\mathbf{N}^{Q_0}$. Let $\mathtt{a}_{Q,\mathbf{d}}(q)\in\mathbf{Z}[q]$ be Kac's polynomial, counting i…

Let $Q$ be a quiver, let $\mathbf{r}\in\mathbb{Z}_{\geq0}^{Q_0}$ be a rank vector, and let $\alpha\geq1$. Higher-depth positivity conjecture. … The conjecture extends positivity fr…

Let $Q$ be a quiver and let $\mathbf{d}\in\mathbf{N}^{Q_0}$ be a dimension vector. The polynomial $C^{\mathrm{abs}}_{Q,\mathbf{d}}(q)$ is the absolutely cuspidal polynomial associa…

Let $\Gamma=(\Gamma_0,\Gamma_1)$ be a finite quiver with no loops, let $\alpha\in\mathbb{N}^n$ be a dimension vector, and let $A_\Gamma(\alpha,q)$ be its Kac polynomial. Let the Ka…

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…

Stabilization conjecture. The sequence

Valuation conjecture. Provided $A_{Q_{\underline{n}},\operatorname{\mathbf{d}}}(q)\neq 0$, its valuation is

Let $g$ and $r$ be integers, let $C_{g,r}^{\mathrm{abs}}$ be the absolutely cuspidal polynomial for curves of genus $g$ and rank $r$, let $GSp(2g,\overline{\mathbb{Q}_l})$ be the g…

Let $\Gamma$ be a quiver with vertex set $\Gamma_0$, let $\R \Gamma_0$ be the associated real vector space, let $\N \Gamma_0$ be the monoid of dimension vectors, and let…

Let $A_{g,r,d}(0)$ be the constant term of the polynomial counting absolutely indecomposable vector bundles of rank $r$ and degree $d$, and let $\mathfrak{h}^{\mathrm{sph}}$ be the…