4 problems

Let $\sigma \\: \Gamma \to 2^V$ be a quasi-geometric subdivision of a simplex, with local $h$-vector $\ell(\Gamma)=(\ell_0,\dotsc,\ell_d)$. Stanley's conjecture. The local $h$-vect…

Chan–Stanley unimodality conjecture. All local $h$-vectors of quasi-geometric subdivisions are unimodal.

Athanasiadis's local gamma-vector conjecture. The local $\gamma$-vector of a flag vertex-induced homology subdivision of a simplex is nonnegative.

Let $V$ be a $d$-element set, and let $\Gamma$ be a flag vertex-induced homology subdivision of the simplex $2^V$. The local $h$-polynomial has a unique expansion … and…