7 problems

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 $\lambda_1,\dots,\lambda_n\in\mathbb{R}_{>0}$…

Self-dual realizability conjecture. All $4$-dimensional strongly involutive combinatorially self-dual polyhedral cones have self-dual realizations.

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 $X$ be an orthogonal Shimura variety arising from an even unimodular lattice of signature $(b,2)$, where $b>2$. The cone of special cycles of codimension $2$ on $X$ is the cone…

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…