18 problems

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…

Let $G$ be a discrete group acting on a $C^*$-algebra $A$ by automorphisms of $C^*$-algebras. For each $n\in\mathbb{Z}$, let the $l^1$-assembly map and the reduced $C^*$-assembly m…

Let $G$ be a topological group with a universal proper space $\mathcal E G$. Let $A$ be a $C^*$-algebra and let $B$ be a $G$-algebra. Define the naive topological $KK$-groups by ……

Farrell–Jones and Baum–Connes assembly conjectures. The maps $\operatorname{asmb}_i$ for $i=1,2,3$ are bijective. These comprise the rational and real assembly maps associated with…

Let $\Gamma$ be a countable discrete group. Let $E\Gamma$ be the universal space for proper and free action of $\Gamma$, and let $\underline{E}\Gamma$ be the universal space for pr…

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…

Let $G$ be a semisimple Lie group of real rank one, and consider its spherical representations in a closed strip. Julg's representation-theoretical conjecture. Spherical representa…

Let $\Gamma$ be a finitely generated discrete group acting on a metrizable finite-dimensional compact space $X$, and let ${\mathcal G}=X\rtimes\Gamma$. Write $RK_i(\Gamma,X)$ for t…

Let $X$ be a discrete proper metric space. The coarse assembly map is … where $K_{*}(P_{d}(X))$ denotes the $K$-homology of the Rips complex of $X$ at scale $d$, and…

Let $\mathcal{G}$ be the groupoid under consideration, with $\widehat{\mathcal{G}}$ its blow-up groupoid and $H_{**}(\widehat{\mathcal{G}};\mathbb{C}_{\widehat{X}})$ the associated…

Let $\mathcal{G}$ be a second countable, étale groupoid, and let $\widehat{\mathcal{G}}$ be the blow-up groupoid with base-related notation $\widehat{X}$. Suppose that $c$-…

Let $\mathcal{G}$ be a second countable, locally compact, Hausdorff, étale groupoid, and let $\widehat{\mathcal{G}}$ be its blow-up at torsion elements of isotropy groups. Write…

Zeidler's conjecture. This non-vanishing still holds without assuming the Baum--Connes injectivity.

The $\tau$-Baum--Connes conjecture with coefficients. The map $\mu_\tau$ is bijective. This is the $\tau$-localised form of the Baum--Connes conjecture with coefficients, asserting…

Integrality conjecture. The morphism $\operatorname{Sign}_{M,\mathcal F;\Gamma}^{\perp}$ is integral. This is motivated by the known integrality of the morphism on the range of the…

Let $(M,\mathcal F)$ be a foliated manifold, let $G$ be its monodromy groupoid, and let $C^*G$ denote the maximal $C^*$-algebra of $G$. For a Galois covering…

Let $Y\subseteq X$ be aspherical compact spaces, and let … be the maximal relative Baum–Connes map. If $j\colon \pi_1(Y)\to \pi_1(X)$ is injective, let … be the reduced relative Ba…

Let $\Gamma$ be a group. Let $K$ be the $C^*$-algebra of all compact operators on an infinite-dimensional separable Hilbert space, and let $C_r^*(\Gamma)$ be the reduced group…