5 problems

Characteristic-polynomial equality conjecture. The arrangements $\mathcal{S}(G)$ and $\mathcal{I}(G)$ have the same characteristic polynomial for any digraph $G$. This conjecture e…

Shi-region low-element conjecture. One has $L_{\operatorname{Shi}}(W,S)=L(W,S)$. Moreover, every Shi region contains a unique low element, which is its unique element of minimal le…

The ideal-arrangement conjecture. All ideal-subarrangements ${\mathcal{A}}_I$ and ideal-Shi arrangements $\operatorname{Shi}^k_{\pm I}$ are divisionally free.

Let $W$ be a crystallographic reflection group, let $\mathsf{Shi}^{(m)}(W)$ be the extended Shi arrangement, and let $\operatorname{coh}(R)$ be the coheight of a positive region…

Let $W$ be a crystallographic reflection group, and let $\operatorname{Cat}^{(m)}(W,q)$ be the combinatorial $q$-Fuss-Catalan number obtained by summing $q^{\operatorname{coh}(R)}$…