9 problems

Let $h=(1,h_1,\ldots,h_e)$ be the $h$-vector of a simplicial polytope. If $h$ is palindromic and unimodal, define its differential sequence by … An $O$-sequence is the Hilbert func…

Let $\Delta$ be an orientable simplicial $\Bbbk$-homology $d$-manifold without boundary, and let…

Let $\Delta$ be a simplicial $d$-sphere, or more generally a simplicial homology $d$-sphere. The $h$-vector of $\Delta$ is required to satisfy the Dehn--Sommerville equations, and…

Let $\Delta$ be a balanced, orientable $\mathbf{k}$-homology manifold without boundary. Define $g”_j(\Delta)=h”_j(\Delta)-h”_{j-1}(\Delta)$. Balanced manifold $g”$-conjecture. The…

Let $\Delta$ be a simplicial (homology) sphere, with $h$-vector and $g$-vector as in the $g$-theorem. The $g$-conjecture for spheres. The conditions of the $g$-theorem should hold…

The g-conjecture. The conditions on the Hilbert function mentioned in the source characterize exactly the Hilbert functions of the Stanley–Reisner rings of triangulations of sphere…

Let $\Delta$ be a homology manifold, let $h'_i$ denote its h-prime vector, and let $\beta_{i-1}$ denote its $(i-1)$st Betti number. Manifold g-conjecture. For every…

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…