9 problems

Cellular replacement conjecture. The labelled regular cell complex $Z$ can be constructed from $Y$ by replacing each vertex $\mathbf{b}_j$ with a copy $\mathbf{b}_jX$; moreover, th…

Let $S$ be a numerical semigroup, let $R$ be its numerical semigroup algebra, and consider the infinite Apéry resolution of the residue field over $R$. Cellular structure conjectur…

The projective cellular resolution conjecture. If $Y$ is a projective cellular resolution of $A$ as an $A \otimes_{\mathsf k} A^{\operatorname{op}}$-module, then $Y$ is homotopic t…

Let $I$ be a monomial ideal with a minimal resolution supported on a regular CW-complex having the intersection property, meaning that the intersection of any two cells is also a c…

Let $A$ be the consistent toric algebra associated to a collection on a Gorenstein affine toric variety $X$ of dimension $n$. Let $\mathbb{T}^n$ be the quotient torus in which the…

Linear-width conjecture. There is a constant $C_d$ such that

A subdivision of a dilated simplex gives rise to a collection of types, and an abstract oriented matroid is an oriented matroid specified abstractly by its axioms rather than obtai…

Let $P$ be a polytope supporting a Cohen–Macaulay cellular resolution. Selfduality conjecture. The polytope $P$ must be selfdual. The preceding lemma establishes that the $f$-vecto…

Let an $n$-gon have a subdivision with $k$ additional edges and no additional vertices. A CM monomial labelling is a monomial labelling whose cellular complex gives a Cohen–Macaula…