112 problems

Let $d\geq 1$, let $G$ be a graph, and let $F$ be a set of nonedges of $G$. For each $f\in F$, write $(G\cup(F\setminus\{f\},f))$ for the graph-nonedge pair obtained by adding the…

Let $\mathcal{H}_g^{V(r)}$ be the small Heisenberg homology, let $\mathcal{H}_g^{V,\dagger}$ be the direct sum of the standard twisted homology groups, and let…

Non-termination conjecture. The LCS of ${\mathbf{B}}_{2,m}({\mathbb{P}^2})$ does not stop.

Homological stability conjecture. The cohomology group

Let $A$ be a $1$-connected Poincaré duality CDGA of formal dimension $n$ odd. For each $\xi\in (A\otimes A)^{2n-2}$, let $C(\xi)$ be the associated CDGA, and let … be the natural s…

Let $\Gamma$ be a planar graph, and let $B_n\Gamma$ and $P B_n\Gamma$ denote its $n$-strand braid group and pure braid group, respectively. A group is right-angled Artin if it admi…

Sundaram–Welker's conjecture. For every number partition $\lambda$,

Let $q_n$ denote the polynomial equation problem associated with the unordered configuration space of $n$ points, let $g(q_n)$ be its Schwartz genus, and let $L_n$ be the relevant…

Atiyah's conjecture. The polynomials $p_1,\ldots,p_N$ are linearly independent. Equivalently, their $N\times N$ coefficient matrix is nonsingular. Atiyah's conjecture is known for…

For $n\ge 2$, let $MT(n)$ be the set of isomorphism classes of meta-trees with $n$ tails, and define a partial order $<$ on $MT(n)$ by declaring ${\bf T}<{\bf T}^{\prime}$ when the…

For $n\ge 2$, let $FM_2(n)$ be the Fulton–MacPherson space, let $MT(n)$ be the set of isomorphism classes of meta-trees with $n$ tails, and let $P$ be the dg-operad whose standard…

Presentation conjecture. The homomorphism $\varphi_n:PL_n\to P_n^2$ is an isomorphism.

Let $\mathcal{X}_{\mathrm{cf}}(3,6)$ be the closure of the variety parametrized by the canonical forms of the real regions of $\mathcal{X}(3,6)$, and let the log canonical compacti…

Stronger twisted coalgebra conjecture. For some model of the $\mathbb{E}_d$ operad, this zig-zag can be promoted to a zig-zag of equivalences

Twisted coalgebra conjecture. This zig-zag can be promoted to a zig-zag of equivalences

Let $\mathcal N_n(\mathbb R^2)$ be the moduli space of planar graphs with $n$ bounded faces, and let $\operatorname{uConf}_n(\mathbb R^2)$ be the configuration space of $n$ unlabel…

Alpert–Kahle–MacPherson's conjecture. The inclusion of $SF_n(R_{w,h})$ into $F_n(\mathbb{R}^2)$ induces an isomorphism

Let $\Lambda$ be a finite representation type algebra, and let $\widetilde{\mathcal{U}}_\Lambda$ and $\widetilde{\mathcal{M}}_\Lambda$ be the affine varieties associated with…

Let $\mathsf{\Gamma}$ be a graph, let $i$ be a homological degree, and let $\lambda$ be a partition. Write $\mathcal{H}_i(\mathsf{\Gamma})$ for the relevant representation-valued h…

Let $\mathcal{G}$ be a collection of graphs. Say that $\mathcal{G}$ is a set of universal generators in degree $i$ if, for every graph $\mathsf{\Gamma}$, the group…

Let $\mathsf{\Gamma}$ be a graph and let $k\geq 0$. Write $F_k(\mathsf{\Gamma})$ for its $k$-point configuration space. Folklore torsion-freeness conjecture. The space…

Acyclic direction contractad homotopy-equivalence conjecture. The map $\psi_d$ induces a homotopy equivalence of topological graphical collections

Let $G$ be a non-oriented finite simple graph, let $C_G(\mathbb{R}^3)$ be its configuration space, and let $\mathcal{A}_G$ denote the associated $G$-amplitude function. Graph-ampli…

Discrete configuration-space conjecture. For all $m\leq n$, the ordered configuration space of $m$ points in $K(n)$ is homotopy equivalent to the $m$th discrete ordered configurati…

Let $F(\mathbb{R}^2,n)$ denote the ordered configuration space of $n$ distinct points in the plane, and let $\mathrm{TC}^{\Sigma_s}[F(\mathbb{R}^2,s+1)\to F(\mathbb{R}^2,s)]$ denot…