17 problems

Lattice-isomorphism conjecture. These lattices are isomorphic, compatibly with the identification predicted by the periodicity conjecture for the corresponding Grothendieck-group c…

Periodicity conjecture. Under this isomorphism,

Let $\Pi_n$ be the poset of set partitions of $\{1,2,\dots,n\}$ ordered by refinement, and let $S\subseteq\{1,2,\dots,n-2\}$. Write $\beta_S(\Pi_n)$ for the rank-selected homology…

For a positive integer $n$, let $C_n(\operatorname{GL}_r(k))$ denote the variety of commuting $n$-tuples in $\operatorname{GL}_r(k)$. Motivic stability conjecture. The limit … exis…

Braid-matroid primitive onset conjecture. The onset of representation stability is $n=2i$ for fixed $i eggeq 1$ when $A(n)=cmathrm{OS}(cmathrm{Br}n)$, and is $n=2i$ for fixed…

Uniform-matroid onset conjecture. For fixed $i eggeq 3$, the sequence $cmathcal{L}(n)i$ is representation stable past $n=2i-1$.

Katada's conjecture. For $n\gg *$,

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…

The hypertoric intersection-cohomology conjecture. For all $n$, there exists an isomorphism of graded $\mathfrak{S}_n$-representations

Let $j$ and $w$ be integers, let $\operatorname{config}(n,w)$ denote the relevant configuration space, and let $\operatorname{FI}_d$ be the category of finite sets and injections w…

For a prime $p$ and a positive integer $k$, consider the irreducible complex representations of $GL_n(\mathbb{Z}/p^k\mathbb{Z})$ for all $n$, together with a compatible collection…

A $VIC(\mathbb{Z})$-module is a functor from the category $VIC(\mathbb{Z})$; it is finitely generated if it is generated by finitely many elements under the morphisms of this categ…

Steinberg-module stability conjecture. For each $i,j\geq 0$, the $\mathrm{VB}_{\mathbb{Z}}/\Gamma(p)$-module

Let $\Pi_n$ be the partition lattice, let $S\subset\{1,2,\ldots,n-2\}$, and set $i=\max(S)$. Let $\beta_S(\Pi_n)$ denote the rank-selected homology $S_n$-representation. Sharp stab…

Let $\mathcal{I}_{g,1}$ be the Torelli group of a genus-$g$ surface with one boundary component, and let $\operatorname{gr}(\mathcal{I}_{g,1})$ be the graded rational Lie algebra a…

Let $P\Sigma_n$ be the pure string motion group and let $W_n$ be the hyperoctahedral group acting on it by permuting labels and reversing orientations. For each fixed $i\geq 0$, co…