11 problems

Let $\cone_k$ denote the modular cone associated with weight $k$. Polyhedrality conjecture. Suppose that $k\equiv 2$ modulo $4$ and $k>4$. The cone $\cone_k$ is polyhedral. The pap…

Let $Q$ be a quiver, let $\beta$ be a Schur root, and let $C(Q,\beta)$ be the associated cone. Proposition … is minimal; equivalently, the facets of $C(Q,\beta)$ are in one-to-one…

Let $L$ be a lattice and let $\sigma$ and $\sigma'$ be cones in a fan, represented by generators $\sigma\leftrightarrow\{v_1,\ldots,v_k\}$ and…

Bounded-ratio conjecture. The optimal bounding constant for any normalized reduced bounded ratio on $\mathbb{L}_n^+$ is at most $2$.

Finite-generation conjecture. The cone $C$ is $(R,G)$-finitely generated if and only if there exists $A\in\mathbb{Q}^{n\times n}$ and $\lambda_1,\dots,\lambda_n\in\mathbb{R}_{>0}$…

Facet characterization conjecture. If $|N|\geq 3$, then the facet-defining inequalities for the cone $E(N)$ are exactly the inequalities corresponding to non-trivial irreducible mi…

Dense periodicity conjecture. If $\mathcal{P}(T)$ is dense in $X$, then $T$ is periodic: there is a positive integer $k$ such that $X=\mathcal{P}_k(T)$.

Let $\mathcal{W}$ be a finite reflection group acting on $\mathbb{R}^n$, with fundamental chamber $\mathcal{C}$ and characteristic polynomial $\chi(t)$ of its associated reflection…

Let ${\cal P}(m,n)$ denote the class of $n$-dimensional polyhedral cones with $m$ extreme rays, and let $\gamma(K)$ be the maximum exponent of a $K$-primitive matrix. Conjecture 7.…

Finite asymptotic-nullity characterization conjecture. Given any $n$, there exists a finite list of polynomial matrices $P_1,\ldots,P_\ell$ such that a homogeneous ratio…

Let $\frac{\beta}{\alpha}$ be a ratio of products of principal minors, and let $\log\left(\frac{\alpha}{\beta}\right)$ denote its formal logarithm. Write $\mathcal B_n$ for the sem…