14 problems

Multiplicity conjecture. The inequalities

Existence conjecture. The following conditions are equivalent:

Let $\mathcal{K}$ be a flag complex, let $\mathbf{a},\mathbf{b}\in\mathbb{Z}_{\geq 0}^{\mathcal{V}}$ have disjoint supports, and write $|\mathbf{b}|$ for the sum of the entries of…

Let $r=\binom{n+1}{2}$, and let $h_{n,0},\ldots,h_{n,r}$ be the nonnegative integers defined by … Set $k=\binom{n-1}{2}$. The $h$-vector conjecture. The coefficients satisfy…

Anisotropy conjecture. For an arbitrary field $\mathbb{F}$, any $\mathbb{F}$-homology sphere is generically anisotropic over $\mathbb{F}$.

Let $L_n^2$ be the graph on $[1,n]$ with edges $\{i,i+1\}$ for $i=1,\ldots,n-1$ and $\{i,i+2\}$ for $i=1,\ldots,n-2$. Let $\Delta_k^t(L_n^2)$ be the associated complex, and let its…

Let $L_n$ be the path graph on vertices $\{x_1,\ldots,x_n\}$, and let $\Delta_k^t(L_n)$ denote the associated complex. The preceding result establishes the claim when $n=2k-1$. Lin…

Let $\bm{a}=(a_1,\ldots,a_m)\in\mathbb{Z}_{>0}^m$ with $d=\sum_{i=1}^m a_i\geq 3$ and $\bm{a}\neq(d-1,1),(1,d-1)$. An $(d-1)$-dimensional simplicial complex $\Delta$ is $\bm{a}$-ba…

Assume $k_1$ is a field of characteristic $2$, $n\geq 1$ is an integer, and $D$ is a simplicial sphere of dimension $n$ with vertex set $\{1,\dots,m\}$. Let $k$ be the field of fra…

Let $I$ be an Artinian monomial ideal, let $J$ be a polarization of $I$, and let $\Delta(J)$ be the associated simplicial complex. Let $k[\Delta(J)]$ denote its Stanley–Reisner rin…

Buchsbaumness conjecture. If the singularities of $\Delta$ are not homologically isolated, then

Vanishing conjecture. If $\operatorname{lk}(f,\mathcal{K})$ is a $4$-gon or a $5$-gon for every codimension-$2$ face $f$, then

Let $\Delta$ be a homology manifold, and let $L^i_s(\Delta)$ denote the set of pairs consisting of a linear system of parameters and a one-form satisfying the partial Lefschetz con…

Let $\Delta$ be a homology sphere. Let $k[\Delta]$ be its Stanley–Reisner ring, let $\Theta$ be a linear system of parameters, and define $L(\Delta)$ to be the set of pairs…