112 problems

Let $X$ be an $F_0$-space, meaning an elliptic space with evenly graded rational cohomology, and let $p \colon E \to B$ be any fibration of simply connected spaces with fibre $X$.…

Rapoport–Zink spectral sequence conjecture. For all $r,w$, the induced map on $E_2$-terms is an isomorphism

Let $L$ be a link in $S^3$. The HOMFLY-PT homologies carry gradings $\operatorname{gr}_T$ and $\operatorname{gr}_M$, and the grading-shifted knot Floer homologies…

Let $K$ be a knot. Its Khovanov homology over $\mathbb{Q}$ is a bigraded vector space, and write $s$ for Rasmussen's invariant. A pawn move piece is … and a knight move piece is ……

Rasmussen's spectral sequence conjecture. There is a spectral sequence from $\overline{Kh}(K)$ to $\widehat{\mathit{HFK}}(K)$.

SKT degeneration conjecture. If $X$ admits an SKT metric, then its Frölicher spectral sequence degenerates at the second page $E_2$.

Dunfield–Gukov–Rasmussen conjecture. There is a spectral sequence from HOMFLY-PT homology to knot Floer homology.

Degeneration conjecture. The spectral sequences degenerate at the $E_3$-term for any reduced plane curve $C:f=0$.

Let $L$ be a link, and let $c$ and $c'$ be components of $L$. Denote by $\widetilde{E}^k(L,c)$ the reduced Szabó spectral sequence associated to $L$ with distinguished component…

Spectral-sequence collapse conjecture. For all $F,G\in\mathcal{P}$, this spectral sequence collapses at the second page. In particular, there is a graded isomorphism, not necessari…

The E2-degeneration conjecture. The spectral sequence degenerates at the $E_2$ page.

The critical-eigenspace conjecture. There is an isomorphism

Let $F$ be a field with $|\cdot F/\cdot F^2|=2^n$, so that $E=E_n\cong (\mathbb Z/2)^n$, and let ${\mathcal G}_F$ be the associated $W$-group. Let…

Let $\lambda\in\mathbb C$, let ${\mathfrak{gl}}(\lambda)$ be embedded in $\mathrm{Dif}_1$, and let … be the first page of the associated Feigin Serre–Hochschild spectral sequence.…

The $G^1$-degeneration conjecture. The spectral sequence $G^k_{p,q}$ for $H_*(X(\mathbb{R}))$ degenerates at the $G^1$ level.

Mokrane's spectral sequence conjecture. For all $r,w$, the map

Spectral sequence differential conjecture. The spectral sequence collapses after the $E_3$ stage, and its differential

Let $\sigma$ be the simplicial resolution of the discriminant $\Sigma$ of the space of long knots, and let … be its natural filtration. Write $\bar\sigma_i$ and $\bar\sigma$ for th…

Vassiliev's stabilization conjecture. The spectral sequence associated with this filtration and computing the Borel–Moore homology groups of the resolution $\sigma$ stabilizes over…

Let $R$ be a ring spectrum and let $E$ be a commutative localized quotient of $R$. For a finite cell $R$-module $M$, let $\hat{\mathrm{L}}^R_E M$ denote the completed $E$-localizat…

Let $\mathrm{X}$ be a holomorphic symplectic manifold, and let $\mathscr{D}_\mathrm{L}$ and $\mathscr{D}_\mathrm{M}$ be quantised orientation modules associated to Lagrangian subma…

Let $Y$ be a closed, oriented $3$-manifold and $K\subset Y$ a knot. Let $HFK'_2(Y,K;\mathbb{C})$ be the homology of the mapping-cone complex … and let $I^\sharp(Y,K;\mathbb{C})$ de…

Let $F$ be a field of low cohomological dimension. Consider the motivic Adams spectral sequence based on $\operatorname{BPGL}\langle 2 \rangle$, and let $n$ be a positive integer.…

Let $\mathrm{HHH}_{S^n}(O_1)$ be the $S^n$-colored triply graded Khovanov–Rozansky homology of the unknot, with even generators $u_0,\ldots,u_{n-1}$ and odd generators…

Let $\phi$ be the flow under consideration, let $G_+$ be its associated graph, and let $SFC(\phi,\mathcal{C})$ and $CA(\phi)$ be the chain complexes described above, where…