5 problems

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…

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…

Let $X$ and $Y$ be u.l.f. metric spaces, and let $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_u(Y)$ denote their uniform Roe algebras. An isomorphism between these algebras is said to ind…

Let $X$ and $Y$ be u.l.f. metric spaces. The uniform Roe algebra rigidity conjecture. If their uniform Roe algebras … are isomorphic, then $X$ and $Y$ are bijectively coarsely equi…

Let $X$ be a bounded geometry metric space, and let $C^*_u(X)$ denote its uniform Roe algebra. The real-rank conjecture. If $X$ has asymptotic dimension at least two, then…