7 problems

Let $c$ be a positive integer. Let $\mathbf{M}$ be a finitely generated $\operatorname{OI}$-module over $\mathbf{P}=({\mathbf{X}}^{\operatorname{OI},1})^{\otimes c}$, or a finitely…

Linear projective-dimension conjecture. The projective dimensions of these modules should be linear in $n$.

Independence-number rationality conjecture. The generating function is rational. In particular, the function $n \mapsto \alpha(G_n)$ agrees with a quasi-polynomial for all…

Let $\FIVAnly$ be the $\infty$-category of analytic $\FI$-objects, and let $\FIVTors$ be the full sub-$\infty$-category of torsion $\FI$-objects. For full subcategories…

Let $a,b\in\mathbb{N}$, let $n$ be a homological degree, and let $\mathfrak{Crit}(a,b)$ denote the set of critical pairs $(\lambda,\mu)$ used in the proposition. For each such pair…

Regularity conjecture.

Let $\bk$ be an arbitrary field. Let $\FI_d$ be the category whose objects are finite sets and whose morphisms are injections with $d$-coloring of the complement: a morphism from…