100 problems

Coarse Baum–Connes conjecture. The assembly map is an isomorphism for every $i$. This conjecture relates the large-scale geometry of $X$ to the $K$-theory of its Roe algebra. The s…

Coarse Novikov conjecture. The induced map

Let $M$ be a uniformly contractible Riemannian manifold with bounded geometry. Gromov's conjecture. $M$ cannot have uniformly positive scalar curvature. The conjecture is presented…

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…

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…

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…

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…

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…

A uniformly contractible Riemannian manifold is considered with its bounded L-theory assembly map. Bounded Farrell–Jones isomorphism conjecture. In dimensions higher than four, the…

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…

Coarse geometric Novikov conjecture. The map $\mu$ is injective. This is the coarse-geometric analogue of the strong Novikov conjecture, formulated for bounded-geometry metric spac…

Coarse Baum–Connes conjecture. The rational Roe index map is an equivariant split monomorphism. This version is described as a further coarsening of the condition $FW$ and as imply…

Weinberger's conjecture. The homomorphism $\delta$ is an epimorphism. This is one of the coarse statements that imply a version of the Novikov conjecture; the supplied text gives n…

A universal Cayley graph would be a Cayley graph into which every finitely generated group coarsely embeds, and a universal quasi-isometry class would be a quasi-isometry class con…

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…

The equivariant coarse Baum--Connes conjecture. The map $\mu_X^{\Gamma}$ is an isomorphism. This conjecture asserts that the large-scale equivariant index-theoretic information of…

Bounded-geometry to $\ell^\infty$-coefficient analysis conjecture. This setup results in a framework mapping bounded geometry to analysis with $\ell^\infty$-coefficients.

Absolute and relative $L^2$-Euler characteristic conjecture. The uniform coarse index of $D^\chi_{\mathrm{abs/rel}}$ computes an appropriate absolute or relative $L^2$-Euler charac…

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 $X$ be a discrete metric space with bounded geometry, let $P_s(X)$ be its Rips complexes, and let $\Gamma(X,\mathcal{A}\otimes\mathcal{K})$ be a coarse $X$-algebra. The twisted…

Let $n$ be a positive integer, and let $A$ be a subspace of $\mathbb{R}^n$. The asymptotic dimension of $A$ is denoted by $\operatorname{asdim}(A)$, and $A$ separates…

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…