31 problems

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…

Prime-power congruence conjecture. One has

The prime-power generalization conjecture. For every integer $k\geq0$, there exists such a polynomial $c_k(t)$ for which

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…

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 … 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

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…

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…

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…

Babai–Vu–Wood conjecture. $\varphi$ is irreducible with high probability.

Flag-geometric characteristic polynomial conjecture. One has

Let $A_i$ denote an antichain of size $i$, and let $P$ be a poset whose Hasse diagram is contained in the Hasse diagram of $A_k\oplus A_2$. Let $M^{P}$ be the promotion matrix of…

Let $R=\{\mathfrak p_1,\ldots,\mathfrak p_r\}$, let $E_R^*$ be the group of totally positive $R$-units, let $\vartheta\in H_{n+r-1}(E_R^*,\mathbf Z)\cong\mathbf Z$ be a generator,…

Let $M_n(\mathbb{C})^{+}$ denote the positive semidefinite $n\times n$ complex matrices, and let a positive contraction be a positive semidefinite matrix bounded above by the ident…

Let $\mathrm{M}$ be a matroid on a finite set $E$, and write $r+1$ for its rank. Let $\chi_{\mathrm{M}}(\lambda)$ be its characteristic polynomial, and let $w_k(\mathrm{M})$ be the…

Let $(\mathcal{G},\ast,0,\leq)$ be an ordered Abelian group, let $\mathcal{S}=\mathcal{G}^{(0)}\cup\mathcal{G}^{(1)}\cup\{\varepsilon\}$ be the associated supertropical semifield,…