12 problems

Let $\Delta$ be a Cohen–Macaulay simplicial complex, and let its $h$-vector provide the prescribed ranks. A Boolean interval in the face poset is an interval isomorphic to a Boolea…

Bjorner--Swartz conjecture. Every doubly Cohen--Macaulay simplicial complex has the dual WLP.

Björner–Swartz conjecture. The $g$-vector of any doubly Cohen–Macaulay complex is an $M$-sequence.

Cohen–Macaulay conjecture. Each individual $E_\pi$ and each union

Sheaf upper-bound conjecture. Then

Upper-bound conjecture. Then

Existence range conjecture. There exists a bi-Cohen–Macaulay simplicial complex $\Delta$ that is not a cone if and only if

Let $A_{2k}$ be the indicated singularity, let $S$ be the base of its miniversal deformation, and let $D(\ell)\subset S$ denote the corresponding Severi stratum. For integers…

Let $\Delta$ be a Cohen–Macaulay simplicial complex, and let $s_1,\ldots,s_n\in\mathbb{N}$. Write $\Delta^{(s_1,\ldots,s_n)}$ for its expansion. Expansion Cohen–Macaulayness conjec…

Consider Cohen–Macaulay flag simplicial complexes and flag simplicial complexes, with $h(\Delta)$ denoting the $h$-vector and $f(\Gamma)$ denoting the $f$-vector. The Cohen–Macaula…

Let $\Delta$ be a $(d-1)$-dimensional simplicial complex that is doubly Cohen–Macaulay over $k$. Write ${\mathfrak h}_i(\Delta)$ for its ${\mathfrak h}$-vector and…

Weak-Lefschetz conjecture for homology spheres. If the intersections $S_i\cap(\bigcup_{j<i}S_j)$ can be taken to be Cohen–Macaulay and the $S_i$ can be taken to be homology spheres…