9 problems

Let $P$ be a Gorenstein IDP polytope, meaning an IDP lattice polytope whose associated Ehrhart ring is Gorenstein. Let $h^*_P(x)$ denote its $h^*$-polynomial. Hibi–Ohsugi conjectur…

Batyrev–Nill polynomiality conjecture. For arbitrary Gorenstein polytopes, the stringy $E$-function is a polynomial in $u$ and $v$. Nill and Schepers subsequently proved this state…

Let $P$ be the Birkhoff polytope, whose points are the real doubly stochastic $n\times n$ matrices, and let $E(P)$ be the affine monoid encoding the magic squares. Question (ii) as…

A Gorenstein simplex is a lattice simplex that is Gorenstein, and a lattice pyramid is a lattice polytope obtained as a lattice pyramid. Fix positive integers $k$ and $v\ge 2$, and…

Let $\Gamma$ be a pure simplicial complex whose facets all have $n$ elements. Let $\mathbf{2}$ denote the binary weight vector, and let $\operatorname{Marg}(\Gamma,\mathbf{2})$ be…

Let $P$ be a Gorenstein polytope of dimension $d$ and degree $s$. A Gorenstein polytope is reducible with factors $F$ and $G$ if it is a Cayley join of $F$ and $G$ and … equivalent…

Let $P$ be a Gorenstein polytope of degree $s$, and let $d(s)$ be a function such that every Gorenstein polytope of dimension at least $d(s)$ is a lattice pyramid. Batyrev–Juny's c…

Let ${\boldsymbol s}=(s_1,s_2,\ldots,s_n)$ be any sequence, and let $\overleftarrow{\boldsymbol s}$ denote its reverse. Let $\mathbf{P}_n^{({\boldsymbol s})}$ be the corresponding…

Let $P$ be an $n$-dimensional Gorenstein polytope of degree $d$. A polytope is a pyramid if it is the lattice pyramid over another lattice polytope. Pyramid conjecture. Every $3d$-…