55 problems

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

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…

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…

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 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…

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…

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…

Coarse Baum–Connes conjecture with coefficients. The homomorphism

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…

Let $X$ be a metric space with bounded geometry, let $P_d(X)$ be its Rips complex at scale $d$, and let $\Gamma(X,\mathcal A)$ be a coarse $X$-algebra. The twisted assembly map is…

Strong Novikov conjecture. The induced map

Coarse Novikov conjecture. The induced map

Quasi $n$-bottlenecking conjecture. If $G$ is coarsely $n$-bottlenecked, then it is quasi-isometric to an $n$-edge bottlenecked graph.

Coarse Menger-type conjecture. If a graph is not coarsely bottlenecked, it must contain $D_n$ as an asymptotic minor for every $n\in\mathbb{N}$.

An $n$-asymptotic Cantor manifold is a metric space of asymptotic dimension $n$ that cannot be coarsely separated by a subset of asymptotic dimension at most $n-2$. For each intege…