12 problems

Generic local-system cohomology conjecture. For every $p$ and almost all $a\in A_1^*$ among those with

Consider the algebra obtained from the Orlik–Solomon algebra by introducing two generators $x_{ij}$ and $y_{ij}$ for each pair $1\leq i\ne j\leq n$ and imposing the analogous relat…

Symmetrised special-involution basis conjecture. For any special involution $\sigma$, the symmetrisation of $\Omega_{\sigma}$ under the action of $G$ is a non-zero element of…

Falk's converse conjecture. If $M$ is line-closed, then $A(M)$ is quadratic.

Line-closed matroid–quadratic Orlik–Solomon conjecture. $G$ is line-closed if and only if $A(G)$ is quadratic.

Let $G$ be a matroid, and let ${\mathcal A}(G)$ be its Orlik-Solomon algebra. The matroid $G$ is line-closed if every line-closed set is closed, where the line-closure of a set…

Let $\mathcal{L}$ be a geometric lattice. Let $\mathrm{OS}(\mathcal{L})$ be its Orlik–Solomon algebra, and let $\mathbb{I}_{\mathcal{L}}$ be the map defined in the source. Quasi-is…

Let $\mathcal{L}$ be a geometric lattice, and let $\mathrm{OS}(\mathcal{L})$ be its Orlik–Solomon algebra. Orlik–Solomon Koszulness conjecture. The algebra…

Orlik–Solomon strong log-concavity conjecture. The Orlik–Solomon algebra $A(M)$ and the reduced Orlik–Solomon algebra $B(M)$ are strongly equivariantly log concave.

Supersolvability conjecture. A matroid (an arrangement) is supersolvable if and only if its Orlik–Solomon algebra is Koszul.

Let ${\mathcal{A}}$ be a hyperplane arrangement, let $A$ be its Orlik–Solomon algebra, and define the first resonance variety by … A neighborly partition is a partition of…

Let $A$ be a hyperplane arrangement in $\mathbb{C}\mathbb{P}^n$, and let $V$ be an $n$-isotropic subspace of its Orlik–Solomon algebra of dimension $d\geq n$ that is not contained…