10 problems

Assume $u\leq d/2$. Let $(\Delta,p)$ be a $(d-1)$-sphere in $S(d-u,d-1)$, with $p$ a generic or natural embedding in $\mathbb{R}^d$. Let $\mathcal{S}^a_i(\Delta,p)$ denote the spac…

Let $\Delta$ be a boolean complex with an action by a group $G$ of automorphisms, and let $\operatorname{Sd}\Delta$ be its barycentric subdivision, with the induced $G$-action. Let…

Let $\Delta$ be a boolean complex, let $\operatorname{Sd}\Delta$ be its barycentric subdivision, and let $\Bbbk[\Delta]$ and $\Bbbk[\operatorname{Sd}\Delta]$ be their face rings. L…

Let $\mathbbm{k}$ be any field of characteristic $p$, let $\mu$ be any $(d-1)$-dimensional cycle over $\mathbbm{k}$, and let $\mathbbm{k}[|\mu|]$ be the associated graded commutati…

Let $\mathbbm{k}$ be any field of characteristic $p$, let $\varSigma$ be a $(d-1)$-dimensional pseudomanifold over $\mathbbm{k}$, and let $\mathbbm{k}[\varSigma]$ be its associated…

Let $\mathbbm{k}$ be any field, let $\varSigma$ be a $(d-1)$-dimensional pseudomanifold over $\mathbbm{k}$, and let $\mathbbm{k}[\varSigma]$ be its associated graded commutative fa…

Let $\Delta$ be a simplicial poset with associated cell complex of dimension $d-1$, let $G$ be a subgroup of $\operatorname{Aut}(\Delta)$, and let…

Let $\Delta$ be an $m$-dimensional complex, and let $\mathbb{C}(\Delta)$ denote its face ring. Let $\omega$ be the one-form used for multiplication in this ring, and write…

Gorenstein quotient conjecture. The ring $\overline{\mathbf{k}[\Delta]}$ should be Gorenstein. This is presented as a closely related, potentially weaker conjecture to the precedin…

Socle characterization conjecture. The equality