16 problems

Bohn–Cameron–Müller's conjecture. The group $G_{K,y}(T_M)$ is maximal, i.e. it is isomorphic to the symmetric group on $\operatorname{r}(M)$ letters:

Let ${\mathcal{A}}$ be a rank $r$ central multiarrangement in $\mathbb{C}^n$ with $|{\mathcal{A}}|=m$. Define … Here $\Omega_n$ is the superspace algebra, $\lambda_L$ is the linear…

Let $g$ be a positive integer, and let $\{M_i\}$ and $\{N_i\}$ be sequences of matroids. One-step separation conjecture. There exist two infinite sequences of matroids $\{M_i\}$ an…

Let $M$ be a non-separable matroid. Let $C$ be a largest circuit of $M$, let $D$ be a largest cocircuit of $M$, and let $\mathcal{B}(M)$ denote the set appearing in the genus bound…

Let $\mathcal H$ be a hypergraph and let $h$ be a hypertree. Define the order $<^{\mathrm{violet'}}_h$ from the tour of a violet Jaeger tree, and use this order to define embedding…

Let $T(K_n;a,b)$ denote the Tutte polynomial of the complete graph $K_n$, and let $p$ be an odd prime and $k$ a positive integer. Let $u$ denote congruence modulo the indicated po…

Flag-geometric characteristic polynomial conjecture. One has

Cut-at-vertex exterior-polynomial conjecture. For every such $\mathcal H$, $\tilde{X}_{\mathcal H}=X_{\mathcal H}$, $\tilde{X}_{\mathcal H}$ is independent of the ribbon structure,…

Conjectured convolution-inverse formula. For the relevant indices $a,b,c,d$,

Let $w\in S_n$. Let $DM_w$ be the diagram matroid associated to the permutation diagram of $w$, and let $\mathscr{P}_w$ be the corresponding positroid. Tutte-polynomial equality co…

Semistable Tutte polynomial positivity conjecture. One has $T_Z(Q;t,q)\in\mathbb{N}[t,q]$. The conjecture is motivated by the positivity theorem for $a_Z(Q)$, and the source sugges…

Let $G=(V_0,V_1,E)$ be a connected bipartite graph inducing hypergraphs $d4a2_0$ and $d4a2_1$ by taking the neighborhoods of the vertices in $V_0$ and $V_1$, respectively. The inte…

Let $M$ be a finite connected matroid with positive rank $n=r(M)$, and let $K$ be a field of characteristic zero. The symmetric Galois group conjecture. The Galois group of the Tut…

Zero-density conjecture. As $G$ ranges over all graphs, the zeros of $Z_G(q,v)$ are dense in regions (a)--(e). More precisely, for each fixed $q$ the zeros are dense in $v$, or for…

Sign-variation conjecture. Fix real $q$ and $v$ in one of these regions. Then, for all sufficiently large $n$ (depending on $q$ and $v$) and all sufficiently large $m$ (depending o…

Zero-free interval conjecture. Suppose $0<q<1$. Then there exists a strictly increasing sequence of self-dual intervals ${\mathcal{V}}_m(q)$, $m\geq 2$, such that