253 problems

Tree discriminant magnitude homology conjecture. For $k=3,4$,

For every annular link $L$, the annular Khovanov homology of $L$ is nonzero in the $k$-grading equal to the wrapping number of $L$:…

For every finite-dimensional associative algebra $A$ over a field $K$, the quotient of its Hochschild cohomology ring by the ideal of nilpotent elements is a finitely generated $K$…

Sazdanovic–Scofield conjecture. The homological span satisfies

Generic cohomology conjecture. For all $F,G\in\mathcal{P}$, there is a graded isomorphism, natural in $F$ and $G$,

Derivation-module dimension conjecture. If

Sum's conjecture.

Extension-group conjecture. For $a,b\in Q$, one has

Conjectural Poincaré–Betti series formula. The polynomial $q_n(t)$ satisfies $q_n(-1)\neq 0$, and

Tsygan's generalized Gerstenhaber formality conjecture. For any associative algebra $A$, the Hochschild chain complex $C_{\bullet}(A,A)$ is a $G_{\infty}$ module over the…

Kontsevich–Soibelman conjecture. For any saturated DG algebra $A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ over a field $K$ of characteristic $0$, the Hodge-to-de Rham spectral seq…

Inverse-limit comparison conjecture. The map above is an isomorphism of the cohomology groups.

Comparison conjecture. There is an isomorphism of homology theories

Contraction conjecture. The contraction $C_{13}$ induces an isomorphism

Let $A$ be the algebra governing the principal block of category $\mathcal{O}$, let $\Delta$ be the direct sum of the relevant Verma modules, and let $\mathtt{E}$ be the graded ext…

Let $\mathfrak{g}$ be a semi-simple complex finite-dimensional Lie algebra, let $W$ be its Weyl group with longest element $w_0$, and let $\mathfrak{t}(w)$ and $\mathfrak{i}(w)$ de…

Let $R$ be a local ring and let $F$ be a differential $R$-module admitting a finite free flag. Suppose that $\operatorname{H}(F)$ has non-zero finite length, and write $d=\dim R$.…

Let $U_n=U(L_n)$, where $L_n$ is the quadratic Lie algebra generated by the elements $\mu_{ijk}$. Consider the natural morphisms $U_{n+1}\to U_n$ and $L_{n+1}\to L_n$ sending…

Bar-complex resolution conjecture. If $\mathcal{P}$ is a Koszul operad, then ${{\mathsf{B}}_{\bullet}^{\mathcal{P}}\!A}$ is a resolution of the left $A$-module…

Let $G$ be a compact $p$-adic analytic group, let $H$ be a closed normal subgroup of $G$, and write $\Omega_{G,H}$ and $\Lambda_{G,H}$ for the corresponding Iwasawa algebras. For…

Bockstein conjecture. For $N$ even,

Koszul bar-to-levelled conjecture. The composition

Let $d=d_2+d_k$ be a codifferential, and let $d_e=d_2+d_k+d_l+\cdots$ and $d_e'=d_2+d_k+d_l'+\cdots$ be extensions. Two codifferentials are formally equivalent when a formal automo…

Let $k$ be a field and let $T$ be a $k$-Tannakian Segal category. Let $\mathcal{H}(T)$ denote a $k$-Tannakian category and let $Ind(\mathcal{H}(T))$ be its ind-category with its in…

Let $B$ be a Noetherian algebra, and let $N$ be a Wedderburn projective $B$-module. Write $\tau_N(B)$ for the trace of $N$ in $B$. Wedderburn-projective Idempotent Nakayama Conject…