33 problems

Church–Farb representation stability conjecture. The sequence induced in cohomology and homology by this stabilization sequence satisfies representation stability.

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

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 $\Pi_n$ 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…

Let $d$ be a positive integer, and for each $n$ let … These form graded $\mathfrak{S}_n$-representations and, with the additional structure described in the source,…

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…

Let $F_n$ be the free group of rank $n$, let $\operatorname{IA}_n$ be the kernel of the action of $\operatorname{Aut}(F_n)$ on $H_1(F_n,\bZ)$, and consider the short exact sequence…

Let the lines of the intrinsic hyperplane arrangement under consideration be classified as stable or unstable according to the stability notion used in the paper. Abundance conject…

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…

Braid-matroid onset conjecture. For every $i eggeq 0$, the sequences $cmathrm{OS}(cmathrm{Br}n)^!i$ and $cmathrm{VG}(cmathrm{Br}n)^!i$ start to stabilize exactly when $n=4i$ and…

Onset-of-stability conjecture. The onset of stability for $cmathcal{L}(n)_i$ is at $2i$ for high enough $i$.

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 $\mathcal{H}^\Gamma_{d,C}(\mathcal{X},\mathcal{L})$ be the twisted analogs of mapping stacks over configuration spaces $\operatorname{Conf}_\Gamma(C)$, equipped with an action…

Strong equivariant log-concavity conjecture. For all $n$, the graded $\mathfrak{S}_n$-representations $A_n$, $B_n$, $C_n$, $D_n$, and $M_n$ are all strongly equivariantly log conca…

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…

Rhoades–Wilson conjecture. The coefficient of $z^{n-k}$ in the graded Frobenius characteristic satisfies

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…

Let $\{\lambda_i\}$ be a nested sequence of Young diagrams such that … with $|\lambda_i|=i$. Let $\tau_{k+1}=(k+1,k,\ldots,1)$, and define the $k$-stable shape of $\{\lambda_i\}$ b…

Let $\{\lambda_n\}$ be a nested sequence of Young diagrams with $|\lambda_n|=n$, and let $p_k(n)$ denote the eventual dimension polynomial for the degree-$k$ part of the Springer r…

Let ${{\mathcal I}}_g^b$ be the Torelli group of a genus-$g$ surface with $b$ boundary components, and let $\operatorname{H}_k({{\mathcal I}}_g^b)$ denote its $k$th homology group.…

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