56 problems

For every finitely generated commutative monoid $M$ and every bounded-to-one Borel action $M\curvearrowright X$, the action has finite Borel asymptotic dimension:…

Coarse Baum–Connes conjecture with coefficients. The homomorphism

Strong Novikov conjecture. The induced map

Fixed-radius coarse separator strengthening. For every $k,r\in\mathbb{N}$, there exists $k'\in\mathbb{N}$ such that if $G$ admits $(k,r)$-balanced separators, then $G$ admits a…

Coarse Baum–Connes and coarse Novikov conjectures. The coarse Baum–Connes conjecture claims that $\mu$ is an isomorphism, while the coarse Novikov conjecture claims that $\mu$ is i…

Let $\Gamma$ be a countable discrete group acting properly on a proper metric space $X$. Let $KX_*^{\Gamma}(X)$ denote the universal $K^{\Gamma}$-group assembling the equivariant i…

Equivariant coarse Novikov conjecture. The map $\mu_X^{\Gamma}$ is rationally injective. This is the injectivity part of the equivariant coarse Baum–Connes conjecture and implies t…

Coarse Novikov conjecture. The induced map

Bounded-distance crossings conjecture. There is an integer $k$ such that for every pair of crossing edges $uv,xy$ in $G$, we have

Rough Baum–Connes conjecture. The map $\mu_u$ is an isomorphism. This conjecture asserts that uniform $K$-homology of the Rips complexes computes the $K$-theory of the uniform Roe…

Boundary coarse Baum–Connes conjecture. This assembly map is an isomorphism. This conjecture concerns the boundary version of the coarse Baum–Connes conjecture and is used in appli…

The coarse geometric Novikov conjecture. The map $Ind$ is injective.

Abrishami–Czyżewska–Kluk–Pilipczuk–Pilipczuk–Rzążewski's conjecture. For every $k,r\in\mathbb{N}$, there exist $k',r'\in\mathbb{N}$ such that if $G$ admits $(k,r)$-balanced separat…

A length space is a metric space in which the distance between two points is the infimum of the lengths of paths joining them. A length space is quasi-isometric to a graph when the…

Uniform Roe algebra rigidity conjecture. The $C^*$-algebras $C^*_u(X)$ and $C^*_u(Y)$ are Morita equivalent (respectively, isomorphic) if and only if $(X,d)$ and $(Y,\partial)$ are…

Relative coarse Baum–Connes and relative coarse Novikov conjectures. The relative coarse Baum–Connes conjecture for $(X,Y)$ claims that $\mu_{Y,\infty}$ is an isomorphism, while th…

Let $\Gamma$ be a discrete countable group with a length function $|\cdot|$, let $P_d(\Gamma)$ be its Rips complex at scale $d$, and let $C^{\ast}(P_d(\Gamma),A)^{\Gamma}$ and…

Drutu–Nowak conjecture. The map $\mu_c$ is not surjective for $\mathcal O_\Gamma M$. The conjecture makes precise Roe’s proposed use of warped cones to produce counterexamples to t…

Analytic equivariant coarse Novikov conjecture. The composition of $\pi_*$ and the assembly map, namely the Miščenko–Kasparov assembly map $\nu_X^{\Gamma}$, is injective. This is a…

The $\ell^p$-coarse Baum–Connes conjecture. The induced map $\operatorname{ev}_*$ is a K-theoretic isomorphism. This is the $\ell^p$ localization-algebra formulation of the conject…

Uniformly quantitative coarse Baum–Connes conjecture with coefficients. For every $k\in\mathbb{N}$, there exists $0<\varepsilon<1/4$ such that, for every $r>0$, there exists…

Quantitative coarse Baum–Connes conjecture with coefficients. For every $k\in\mathbb{N}$, there exists $0<\varepsilon<1/4$ such that, for every $r>0$, there exists $k'\geq k$ for w…

Let $A$ be a filtered $C^{\ast}$-algebra, let $X$ and $Y$ be proper metric spaces, and let $N_X$ be a locally finite net in $X$. For each $k\geq 0$, let $P_k(N_X)$ be the $k$-Rips…

Let $A$ be a filtered $C^{\ast}$-algebra, let $X$ be a proper metric space, and let $N_X$ be a locally finite net in $X$. Let $\bigsqcup_{\mathbb{N}}X_i$ denote the separated coars…

Let $X$ be a metric space with bounded geometry and let $\Gamma(X,\mathcal A)$ be a coarse $X$-algebra. Denote by … the twisted assembly map associated with $X$ and $\mathcal A$. T…