9 problems

Characteristic-independence conjecture. An $h$-vector is level in some characteristic if and only if it is level in every characteristic.

Zanello's conjecture. If $R/I$ is level of codimension three, then

Let $R=k[x_1,\ldots,x_n]$ and let ${\mathfrak c}\subset R$ be a complete intersection generated by forms all of the same degree. Let $A$ be a relatively compressed level Artinian q…

Let $\Delta$ be a triangulation of a sphere, and let $P_\Delta$ be its face poset, consisting of all faces of $\Delta$, including $\emptyset$, ordered by inclusion. Recall that…

Let ${\boldsymbol s}\in\mathbb{Z}_{\geq 1}^n$ be a sequence. Suppose there exists ${\boldsymbol c}\in\mathbb{Z}^n$ such that … for every $j>1$, with $c_1=1$. Kohl–Olsen's levelness…

Let ${\boldsymbol s}\in\mathbb{Z}_{\geq 1}^n$ be a sequence, and let $\mathbf{P}_n^{({\boldsymbol s})}$ denote the corresponding $s$-lecture hall polytope. Levelness conjecture. Su…

Consider the question whether every level Hilbert function satisfying the condition referred to as Question (b) in the source is unimodal. Non-unimodality conjecture. The answer to…

For positive integers $e$ and $t$, consider level Hilbert functions of codimension $r$, socle degree $e$, and type $t$. Asymptotic unimodality conjecture. For fixed positive intege…

Let $(1,h_1,\ldots,h_e)$ be an $h$-vector of an artinian level algebra. For a positive integer $\alpha$, suppose that both … and … are level $h$-vectors. Interval Conjecture. Then…