6 problems

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 ${\boldsymbol s}\in\mathbb{Z}_{\geq 1}^n$ 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…

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

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…