49 problems

Let $\Phi$ be an irreducible root system with Coxeter number $h$, and let $V$ be its ambient vector space. For integers $a\leq b$, define the truncated affine Weyl arrangement … wh…

Let $R$ be an irreducible root system in $\mathbb{R}^n$, and let $a$ and $b$ be integers with $a+b\geq 2$. Define the truncated $R$-affine arrangement ${\cal A}^{ab}(R)$ by the hyp…

Let $\Phi$ be an irreducible crystallographic root system in $\mathbb R^l$, not necessarily reduced, and fix a system of positive roots $\Phi^+$. For nonnegative integers $a,b$, no…

For a root system $\Delta$, let $\Delta_s^+$ and $\Delta_l^+$ denote its positive short and long roots. The extended semi-Shi arrangement $\text{Shi}_s^m(\Delta)$ consists of the h…

Prime-power congruence conjecture. One has

For a binary matroid $M$, let $\Lambda(M)$ be the minimum, over bases $B$ of its cocycle space, of the maximum size of a cocircuit in $B$. Let $C(M,t)$ denote its characteristic po…

Let $Z_N(\mu)$ be the characteristic polynomial of an $N\times N$ random matrix from the Gaussian Orthogonal Ensemble, and let $\epsilon$ denote $\operatorname{Im}\mu$ scaled by th…

Ellipsoidal separation conjecture. For every fixed $\alpha\in(0,1)$, there exists a dimension $n_0(\alpha)$ such that, for all $n\ge n_0(\alpha)$, the three ellipsoids…

Conjectural characterization. For every $m$ with $3\leqslant m\leqslant n$, at least one of these two inequalities holds. The inequalities express a concavity-type regularity in th…

Let $P_n$ be the path graph on $n$ vertices, let $f_n(\lambda)$ be its characteristic polynomial, and let $R_n$ be the set of roots of $f_n(\lambda)=F_{n+1}$, where $F_n$ is the Fi…

Let $P_n$ be the path graph on $n$ vertices, let … be its characteristic polynomial, and let $F_0=0$, $F_1=1$, with $F_n$ denoting the Fibonacci numbers. For each $n$, let $R_n$ be…

Let $\mathfrak{g}$ be a complex simple Lie algebra, let $V$ be an irreducible $\mathfrak{g}$-module with highest weight $\lambda$, and let $\Pi(\lambda)$ denote the set of weights…

Let $A_{N,\tau}$ be a matrix from the real elliptic ensemble with symmetry parameter $\tau\in(-1,1)$, and let $\mu$ lie in the real bulk interval $(-(1+\tau),1+\tau)$. Let $G$ deno…

Coefficient formula conjecture. The coefficient of $x^{2n-5}$ is

Let $G$ be a reaction network containing the motif … with $Y$ absent from every other complex. Let $p_{(h,\ell)}$ and $p'_{(h,\ell')}$ be the corresponding reduced characteristic p…

For a polynomial $f\in\mathbb{Z}[X]$, let $P_n(H;f)$ denote the number of matrices $A\in\mathcal{M}_n(\mathbb{Z};H)$ with characteristic polynomial $f$. Characteristic-polynomial s…

Square-root conjecture. Then $c_{2k_i}=(a_{k_i})^2$, where $a_{k_i}$ is a polynomial on the Lie algebra.

Finite-fiber conjecture for symmetric tensors. All fibers of $\Phi_{\mathrm{sym}}$ are finite.

Characteristic polynomial fiber-dimension conjecture. The generic fiber of $\Phi$ is equidimensional of dimension $n$.

Let $\mathcal{H}$ be an $r$-tree with $r\geq 3$, and let $H$ be a subgraph induced by some vertex subset. Let $\varphi_H$ denote the matching polynomial of $H$, and let…

Let $G$ be a connected graph, let $\mathcal{A}_G$ be its connected subgraph arrangement, and let $\chi(\mathcal{A}_G)$ denote the characteristic polynomial of this arrangement. Int…

Let $R_n(H;f)$ denote the number of matrices $A\in\mathcal M_n(\mathbb Z;H)$ with characteristic polynomial $f\in\mathbb Z[X]$, where $\mathcal M_n(\mathbb Z;H)$ is the set of…

Let $\phi:\mathfrak{sl}(2,\mathbb C)\to\mathfrak{gl}(V)$ be an irreducible representation on a complex vector space $V$ of dimension $m+1$, with highest weight $m$. Define the char…

Sharpness conjecture. For $e \in \mathbb N$ with $e \geqslant 3$ and $q$ a prime power, there exists $N \in \mathbb N$ such that for all $n \geqslant N$ we have equality in the cor…

Critical CUE conjecture. At the critical relation $s=1/\sqrt{k}$,