16 problems

Baum–Connes conjecture for discrete groups. The analytic assembly map

Let $G$ be a discrete group with torsion, and let … be the trace map, where $C_r^*G$ is the reduced $C^*$-algebra of $G$. Let $S$ be the additive subgroup of $\mathbb Q$ generated…

Let $\Gamma$ be a group, let $X=\underline{E}\Gamma$ be the universal space for proper $\Gamma$-actions, and let $\sigma_s$ be the exponential cocycle used to define the reduced tw…

Let $\Gamma$ be a torsion-free discrete group and let $\sigma$ be a multiplier on $\Gamma$. The twisted Baum-Connes map is … Here $B\Gamma$ is the classifying space of $\Gamma$, an…

Unitary Conjugation Groupoid Conjecture. There exists a natural assembly map

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…

Localised Baum–Connes surjectivity conjecture. The map

Equivariant Baum–Douglas conjecture. Equivariant analytic and geometric $K$-homologies should be equivalent in general; equivalently, the map $BD$ should be an isomorphism.

Uniform coarse Baum–Connes conjecture. The map $\mu_u$ is an isomorphism. This is the discrete Rips-complex formulation of the uniform coarse Baum–Connes conjecture, connecting uni…

Uniform coarse Baum–Connes conjecture. The map $\mu_u$ is an isomorphism. This is an index-theoretic assembly conjecture for uniformly contractible spaces of bounded geometry; the…

Let $G$ be a locally compact group. Two $G$-$C^*$-algebras are equivariantly Morita equivalent when they are related by an equivariant Morita equivalence, and two $C^*$-algebras ar…

Let $G$ be a locally compact group. A $G$-$C^*$-algebra is a $C^*$-algebra equipped with a continuous action of $G$ by $C^*$-algebra automorphisms. For a short exact sequence of…

Let $M$ be a connected, closed, oriented, smooth manifold, let $L_M$ be its Hirzebruch $L$-class, and let $f:M\to B(\pi_1M)$ classify the universal cover. The class $f_*(L_M)$ belo…

Lafforgue's Baum–Connes conjecture. The assembly map $\mu$ is an isomorphism.

Let $\beta$ be a bounding class, let $(G,L)$ be a discrete group equipped with a word-length, and let $H_{\beta,L}(G)$ be the associated rapid decay algebra. Write…

Let $\pi\colon M\to B$ be a smooth fibre bundle of compact manifolds, let $p\colon \widetilde M\to M$ be a normal covering with torsion-free deck-transformation group $\Gamma$, and…