115 problems

Let $\mathbb{F}$ be a skew field. Let $G$ be a torsion-free residually finite group of type $\mathsf{FP}_{n+1}(\mathbb F)$ such that the Hughes-free division ring…

Let $G_1$ and $G_2$ be graphs with the same number of vertices, and let $\Delta$ be the dissimilarity measure obtained from the collections of horizontal homologies partitioned by t…

Let $I^n$ be the $n$-cube, let $2_n$ be the free $\omega$-category generated by an $n$-morphism, and let $G_n$ for $n>0$ be the oriented $n$-globe, namely the free $\omega$-categor…

Let $\mathcal{C}$ be an $\omega$-category and let $I^1$ be the interval in the biclosed monoidal structure on $\omega\operatorname{Cat}$. For either choice of sign, consider the…

Let $\mathcal{C}$ and $\mathcal{D}$ be free $\omega$-categories, let $f,g:\mathcal{C}\to\mathcal{D}$ be homotopic non $1$-contracting $\omega$-functors, and let…

Let $\mathcal{C}$ be a free globular $\omega$-category. For a natural number $n$, let $x_i\in\omega\operatorname{Cat}(I^n,\mathcal{C})^\pm$ and let $\lambda_i$ be natural numbers i…

Let $\mathcal{C}$ be an $\omega$-category, let $I^n$ denote the $n$-dimensional cube, let $A$ and $B$ be composable $n$-morphisms, and let $\widehat{\mathcal{C}}$ denote the modifi…

Let $G$ be a Lie group. Regard $BG$ as the classifying space of the underlying discrete topological group, and let $BG^{\rm top}$ be the classifying space with its usual topology.…

Let ${\cal G}$ be an $\mathbb R$-rational building set. A real subspace $C\in{\cal C}_{\cal G}$ is purely complex when it decomposes as … Write $\mathbb R({\cal G})$ for the real s…

For $g\geq 3$ and $L\geq 2$, let ${\rm Mod}_g[L]$ be the level-$L$ congruence subgroup acting trivially on $H_1(\Sigma_g,\mathbb Z/L\mathbb Z)$. Picard-number-one conjecture for le…

A real algebraic variety is a complex variety $X$ defined over the real numbers, with real part $X(\mathbb{R})$ equal to the fixed-point set of complex conjugation. It is an M-vari…

Sundaram–Welker's acyclicity conjecture. For $\lambda\vdash\mu$ with $\lambda\neq\mu$, the space $X_{\lambda,\mu}$ is ${\mathbb Q}$-acyclic.

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

Gromov's conjecture. Every functorial seminorm on singular homology with real coefficients vanishes for simply connected spaces.

Freeness conjecture. The reduced homology group $\widetilde H_i(M_{m,n})$ is free if and only if $i\geq m-2$.

Following the standard shelling of the order complex of the truncated Boolean algebra $B_n-\{\hat{0},\hat{1}\}$, let $\Delta_n$ be the resulting $h$-complex. Edelman–Reiner conject…

Let $\mathcal{C}$ be a non-contracting $\omega$-category freely generated by a precubical set. Let $M_n^{gl^-}(\mathcal{C})$ denote the degree-$n$ elements of the branching semi-gl…

Let $\mathcal{C}$ be a strict non-contracting globular $\omega$-category. Let $C F^{gl}$ be its formal globular complex and $C F^{\pm}$ its formal branching and merging complexes,…

Let $K$ be a precubical set, let $F(K)$ be the free non-contracting $\omega$-category generated by $K$, and let $\widehat{F(K)}$ denote its Kan completion. Let $C^{gl}$ and…

Let $\mathfrak g$ be the Lie algebra used in the quiver-variety construction, let $\mathbf w^1$ and $\mathbf w^2$ be the two framing data, let $\mathfrak L(\mathbf w^i)$ be the cor…

Level-two homology formula conjecture.

Free-rank conjecture. The rank of the free part of $H_n(R_{(m)})$ is $2^{m-1}$ for $n\geq m-1$, while

Let $g$ be sufficiently large, let $\mathcal{HI}_g^1$ denote the handlebody Torelli group, and let $B_3^{b_i}$ be the subspace defined from the Birman–Craggs–Johnson homomorphism.…

Stable first-homology representation conjecture. For $n\geq 5$, there is an isomorphism of $S_n$-representations

Let $M^{n+1}$ be as in the equivariant setting, with a compact Lie group $G$ acting by isometries and satisfying … A $G$-homology class is represented by a fixed $G$-hypersurface…