30 problems

Unstable Gromov–Lawson–Rosenberg conjecture. The vanishing of $\alpha(M)$ decides whether $M$ admits a metric of positive scalar curvature.

Rational strong Novikov conjecture. The group $\Gamma$ satisfies the rational strong Novikov conjecture if $\mu$ is injective. This is a rational injectivity condition on the assem…

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 $\Gamma$ be a discrete group, let $\underline{E}\Gamma$ be its classifying space for proper actions, let $K_*^\Gamma(\underline{E}\Gamma)$ denote equivariant $K$-homology, and…

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…

Baum–Connes' trace-range conjecture. The range of $\dim_\Gamma$ is contained in $Fin^{-1}(\Gamma)$. This was the original proposed trace-range statement; for torsion-free groups,…

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 ……

Maximal rational strong Novikov conjecture. The group $\Gamma$ satisfies the maximal rational strong Novikov conjecture if $\mu_{\max}$ is injective. The paper notes that the assem…

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…

Random regular Cuntz-algebra conjecture. For $p=2$, or for an odd prime $p$ dividing $r-1$,

Cyclic $K_0$ probability conjecture.

Shifted Bernoulli full-shift probability conjecture.

Cuntz-algebra probability product conjecture. The asymptotic probability that the graph algebra associated to the random $r$-regular multigraph is stably isomorphic to a Cuntz alge…

Let $(X,\phi)$ be a Smale space, and write $R^s(X,\phi)$ and $R^u(X,\phi)$ for its stable and unstable equivalence relations. Define the associated Ruelle algebras by … The assumpt…

For an ample groupoid, meaning an étale groupoid whose unit space is totally disconnected, let groupoid homology denote the homology of the groupoid and let $C^*(G)$ be its associa…

The coarse geometric $\ell^p$-Novikov conjecture. If $\Gamma$ is a discrete metric space with bounded geometry, then the index map $e_*$ is injective.

Stable hypereuclidean conjecture. In this situation, the rational version of Theorem 43 should follow from the injectivity result for the external product maps.

Finite-part assembly conjecture. The group $\widetilde{\mathcal M}_{\gamma_1,\ldots,\gamma_n}$ has rank $n$, and every nonzero element of $K_0^{\mathrm{fin}}(C^*(\Gamma))$ lies out…

Analytic Novikov conjecture. The map $e$ induces an injection

Let $\Sigma$ be a Cantor set with a minimal action of $\mathbb{Z}^p$ that preserves a Borel probability measure $\mu$. Let $\sigma$ be the multiplier on $\mathbb{Z}^p$ associated t…

Trace conjecture.

Let $G$ be a group with torsion elements $g_1,g_2,\ldots,g_k$ of distinct positive orders $d_1,\ldots,d_k$. For each $g_i$, let $p_i$ be the idempotent corresponding to $g_i$, and…

Let $M$ be a compact oriented manifold, let $S(M)$ be its structure group, and let $S_0(M)$ be the subgroup generated by elements $[(f,M')]-[(\psi\circ f,M')]$, where $f:M'\to M$ i…

Let $G$ be a countable group. For an element $g\in G$ of finite order $d$, define … in $\mathbb{Q}G$, and let $K_0^{\mathrm{fin}}(C^*(G))$ be the subgroup of $K_0(C^*(G))$ generate…