502 problems

Let $k$ be an algebraically closed field and let $\left({\mathcal C},\otimes,\mathbf{1}\right)$ be a finite tensor category over $k$. Its cohomology ring is…

Positselski's conjecture. Absolute Galois groups of fields containing a $p$-th root of unity satisfy the Koszul property at $p$.

Diaz–Park sharpness conjecture. For every $n\geq 1$,

Coskun–Woolf's stabilization conjecture. Assume $r\geq 1$. Then for each $i\in\mathbb{Z}_{\geq0}$, there is a constant $N(i)$ depending on $i$ such that for every integer $k\leq i$…

Let $n$ be a positive integer, let $p$ be a prime, and let $\Gamma\subset\operatorname{SL}(n,\mathbb{Z})$ be a congruence subgroup. A Hecke eigenclass is an element…

Let $k$ be a base field and consider the diagram of categories of motives and realization functors described in the paper, with rational coefficients. A morphism of motives is test…

A Hamiltonian fibration is a fibration whose structure group reduces to the Hamiltonian symplectomorphism group of the fiber. For a Hamiltonian fibration $F\hookrightarrow P\to B$,…

Conner conjecture. The orbit space $X/G$ has trivial reduced Čech cohomology.

Let $P_n=\mathbb Z/2[u_1,\ldots,u_n]$ be the polynomial algebra with its unstable $\mathbb A$-module structure, where $\mathbb A$ is the 2-primary Steenrod algebra. Define the cohi…

Gersten's conjecture. The group $H^n(X)$ should be computable from the groups

Let $M$ be a compact $4$-manifold, let $J$ be an almost complex structure on $M$, and let $h^-_J$ denote the dimension of the real vector space of closed $J$-anti-invariant $2$-for…

Stiénon–Xu conjecture. For every transitive Courant algebroid, there is an isomorphism between its standard cohomology and its naive cohomology.

Let $X=\Gamma\backslash G/K$ be a locally symmetric space, where $G$ is a connected semisimple non-compact linear Lie group of real rank $1$. A finite cover of $X$ is a covering sp…

Cohomological special effect conjecture. The system ${\mathcal{L}}$ is special if and only if it is cohomologically special.

Let $k$ be a field and let $({\mathcal C},\otimes,\mathbf{1})$ be a finite tensor category over $k$. Its cohomology ring is … and each object $X\in{\mathcal C}$ has cohomology…

Cohomology-vanishing conjecture. If

Let $X$ be a variety, let $f:X\rightarrow X$ be an $a$-polarized endomorphism, and let $H_X$ be an ample divisor such that, for some $a\in \mathbf{Z}_{>1}$, $f^*H_X\sim aH_X$. Tate…

Tautological generation conjecture. The following assertions hold:

Let $\Gamma$ be a directed graph, let $X(SU(2),\Gamma)$ be its graphical configuration space, and let $\overline{\operatorname{OT}}(\Gamma)$ be the reduced Orlik–Terao algebra asso…

Let $E$ be the quadratic field under consideration, let $K_f$ be the chosen level, let $\mathcal{L}_{\bm{n}}(\mathbb{F})$ be the coefficient system, and let $\mathfrak{m}$ be a non…

Let $E$ be a constructive exceptional bundle on $\mathbb{P}^3$, and let $h^i(E)$ denote the dimension of its $i$-th cohomology group, $\mu(E)$ its slope, and $\chi(E)$ its Euler ch…

Let $G/\mathbb{Q}$ be a connected reductive group, let $X$ be the symmetric space for $G(\mathbb{R})$, and let $K\subset G(\mathbb{A}_f)$ be a neat compact open subgroup. Write…

Let $M$ be a compact $4$-manifold, let $J$ be an almost complex structure on $M$, and let $h^-_J$ denote the dimension of the real vector space of closed $J$-anti-invariant $2$-for…

Kumjian–Pask–Sims conjecture. The cubical and categorical cohomology groups should be isomorphic in all dimensions:

Bonk–Heinonen conjecture. The bound can be chosen independently of $K$; equivalently, there should be a constant depending only on $d$ and $l$ such that