38 problems

Let $i$ and $q$ be non-negative integers, let $k$ be a field, and let $C_i^q$ denote the Kato–Kuzumaki property defined in terms of Milnor $K$-theory of degree $q$. Kato–Kuzumaki's…

Lyubeznik's conjecture.

Kropholler–Mislin conjecture. The invariant $n_G$ is finite if and only if

Let $X=\mathbf H_\mathbb R^n$ be real hyperbolic $n$-space, let $\Gamma<\operatorname{Isom}(X)$ be a discrete group, and let $\delta(\Gamma)$ denote its critical exponent. Consider…

Let $R$ be a commutative ring with unit, and let $G$ be a discrete group without $R$-torsion. The group $G$ has jump cohomology of height $k$ over $R$ if there is an integer…

Resolution conjecture. There exist a compactum $Z$ with $\dim Z\leq n$ and a surjective map $r:Z\longrightarrow X$ such that

Let $X$ be a compactum, let $n\geq 2$ be an integer, and let $\mathcal G$ be a collection of abelian groups such that $\dim_G X\leq n$ for every $G\in\mathcal G$. Let…

Moore's conjecture. Assume that for every nontrivial element $x$ in $\Gamma$, at least one of the following conditions holds: $M1)$ $\langle x\rangle\cap H\neq\\{e\\}$; or $M2)$…

Let $K$ be a field, let $p$ be a prime, and let $K^{(p)}$ be a maximal prime-to-$p$ extension of $K$. Write $K^{(p)}_{ab}$ for the maximal abelian extension of $K^{(p)}$. Bogomolov…

The arbitrary-group acyclic resolution conjecture. There should exist a compactum $Z$ with

Let $G$ be a locally finite group, let $R$ be a nonzero $o(G)$-inverting ring, and set … The cohomological dimension conjecture asserts that … Equivalently, after reducing to finit…

Let $\Gamma$ be a virtually torsion-free group. Its geometric dimension $\operatorname{gdim}(\Gamma)$ is the minimal dimension of a contractible complex admitting a properly discon…

Bhatt–Lurie’s cohomological dimension conjecture. The functor

A word hyperbolic Coxeter group is a Coxeter group that is word hyperbolic, and its virtual cohomological dimension is the cohomological dimension of any torsion-free subgroup of f…

Let $F$ be a quadratic field extension of $F_0$, let $A$ be a central simple $F$-algebra with a unitary involution $\tau$ such that $F_0=\{x\in F\mid\tau(x)=x\}$. Write $SK_1(A)$ f…

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

Let $A$ be an Artin group, and let its dimension be the maximal cardinality of a subset of the standard generating set whose generated subgroup is spherical. Let…

Petrosyan's no-jump conjecture. For groups with no $R$-torsion, no jump can occur in $\operatorname{cd}_R$.

Let $G$ be a semisimple simply connected algebraic group over a perfect field $K$ of cohomological dimension at most $2$. Serre's Conjecture II. Every $G$-torsor over $K$ is trivia…

For a group $Q$, let $K(Q,1)$ be a classifying space, and write ${\sf cat}(K(Q,1))$ for its Lusternik–Schnirelmann category. For a homomorphism $\phi\colon G\to H$, let…

Let $\phi:\Gamma\to\pi$ be a homomorphism of discrete groups. The LS-category $\operatorname{cat}\phi$ is the least integer $k$ such that a classifying-space map $B\Gamma\to B\pi$…

Artin group $K(\pi,1)$ conjecture. Every Artin group satisfies the $K(\pi,1)$ conjecture.

Serre's Conjecture II. If $k$ is a perfect field with $\operatorname{cd}(k)\leq 2$, then

Let $R$ be a Noetherian domain, let $S$ be a solid $R$-algebra, and let $I$ be an ideal of $R$. Solid-algebra persistence conjecture. One has … The conjecture asks whether cohomolo…

Let $\ell$ be a prime and let $F$ be a field with separable $\ell$-dimension $\operatorname{sd}_{\ell}(F)\leq3$. Say that $F$ is Rost $\ell^{\infty}$-divisible when Suslin's conjec…