217 problems

Let $\Sigma$ be the surface under consideration, let $\widetilde{\Sigma}\to\Sigma$ be a finite characteristic cover, and let $v\in {\rm H}_1(\widetilde{\Sigma},\mathbb{Q})$ be a no…

Let $M$ be a closed $3$-manifold, let $S(M)$ denote its Kauffman bracket skein module, and let $HP^{\bullet}_{\#}(M)$ be the Abouzaid–Manolescu homology. Abouzaid–Manolescu dimensi…

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 $\lambda$,

A group $G$ is parafree if it is residually nilpotent and has the same nilpotent quotients as some free group. Baumslag's Parafree Conjecture. Every finitely generated parafree gro…

Let $\Omega_n=X({\tt K}_n)$ be the cycle-free chessboard complex, and let … Here, a space is $\mu_n$-connected when its homotopy groups vanish through dimension $\mu_n$. Best-conne…

Kahle–Newman conjecture. For every finite abelian $p$-group $G$,

Let $T_{3n-2,3n-2}$ be the $(3n-2)\times(3n-2)$ torus grid with its induced $l^1$ metric, and let $\mathrm{VR}(T_{3n-2,3n-2};n)$ be the Vietoris–Rips complex at scale $n$. Write…

Let $B_n$ be the Artin braid group and let $\Gamma_{g,k}$ be the mapping class group of an orientable surface $S_{g,k}$ of genus $g$ with $k$ boundary components. Let $B_{\infty}$…

van Straten–Warmt conjecture. The odd Betti numbers of $J_{\mathcal{R}}$ vanish; equivalently,

Cyclic homology recursion. For every positive integer $n$, one has $H_i(({\mathcal C}^{C_{2n}},D))=0$ when $i$ is odd, so that $A_{2n}(q)$ is a polynomial in $q^2$, and

Kahle's conjecture. With high probability, $H_k(X(G),\Gamma)=0$. This conjecture concerns the threshold for homological vanishing in random clique complexes; the paper proves a fir…

For each $m$, let ${\cal D}^*_m$ be the dual diagram complex equipped with the differential $\partial$, and write $H^1_\partial({\cal D}^*_m)$ for its first cohomology group. The v…

Branching-homology vanishing conjecture. For every integer $n>0$,

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 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 $J$ be a finite subset of the real indices, and let ${\frak g}_{0}^{(J)}$ and ${\frak g}_{-}^{(J)}$ be the subalgebras in the triangular decomposition associated with $J$. For…

Let $D6A4$ be a taut graph, let $D6A4^D4$ be the specified family of subgroups, let $M_H$ denote the corresponding covering spaces, and let $D6A4M$ denote the abelian covering of t…

Homology-pattern conjecture. The displayed computations suggest that the nonzero reduced homology of $\Delta_n^3$ occurs in degree $2n-4$, with the first values having ranks…

Let $\mathcal M_2^{\rm cm}$ be the critically marked moduli space of quadratic rational maps, let $E^{\rm cm}$ be its escape locus, and consider the inclusion…

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…

Let ${\cal G}$ be a building set whose maximal element has the stated forest filtration … The filtration should in fact be the sequence of skeletons of a regular cell complex. This…

Harer's conjecture. The induced map