5 problems

Let $\varPi^M_m$ be the finite poset of pure diagrams between $\pi(m)$ and $\pi(M)$, and let $\Delta(\varPi^M_m)$ be its order complex, whose faces are chains in $\varPi^M_m$. Rega…

Stability conjecture. There is a $k_0$ such that, for all $k>k_0$, there are translations of pure diagrams $\pi_1(k),\ldots,\pi_m(k)$ such that every decomposition of…

Schreyer's characteristic-dependence conjecture. The semigroup of Betti diagrams depends on the characteristic of $k$.

Projective-dimension-two equality conjecture.

Boij–Söderberg convex-hull conjecture. The set $\bar B_{m,M}$ is the convex hull of $\Pi_{m,M}$.