268 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…

Elliott's classification conjecture. Unital separable simple nuclear $C^*$-algebras are classified up to isomorphism by $K$-theoretic invariants.

Let $\Gamma$ be a discrete group and consider the assembly map from the $K$-homology of an appropriate classifying space of $\Gamma$ to the $K$-theory of the reduced $C^*$-algebra…

HK conjecture. For $* = 0,1$, there is an isomorphism

Coates–Sinnott conjecture. If $G$ is abelian, then

Let … be the restriction of the cyclotomic map $\Pi_N^{\circ}$, and let the corresponding Eisenstein ideal act on its source and target. Sharifi's Eisenstein-ideal isomorphism conj…

Let $W$ be a D$p$-brane world-volume embedded in space-time $X$ by $\iota:W\hookrightarrow X$, and let $E\rightarrow W$ be the rank-$q$ Chan–Paton bundle. If $W$ is spin$^c$ with a…

Let $X$ be a smooth, projective, geometrically connected curve of genus $g$ defined over $\mathbb{Q}$. Let $K_4^{(3)}(X)$ denote the third Adams eigenspace of…

Let $G_{\lambda}$ denote the stable Grothendieck polynomial indexed by a partition $\lambda$, and let $G_{(n)}$ and $G_{1^n}$ denote the elements indexed by the one-row and one-col…

Type C K-Stanley expansion conjecture.

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…

Karoubi conjecture. The topological $K$-groups of $A$ are isomorphic to the algebraic $K$-groups of $A\otimes\mathcal{K}$:

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…

Integer-coefficient conjecture. The coefficients $c_{G,\lambda}$ are integers for every graph $G$ and every partition $\lambda$.

Decomposability conjecture. $A$ is decomposable if and only if $K_1(A)$ is torsion free.

Independence conjecture. The series $\mathsf{DT}_{r,k}^{\rm ell}(\mathbb{A}^3,q,t,w;p)$ does not depend on the elliptic parameter $p$.

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…

Lück's modified trace conjecture. The image of $\tau_*$ satisfies

Weinberger–Yu's conjecture. The group $\mathcal M_{\gamma_1,\ldots,\gamma_n}$ has rank $n$, and every nonzero element of it lies outside the image of the assembly map

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…

Let $M$ be as above, let $M^\prime\to M$ be the induced connected normal $\Gamma^\prime$-covering, and let $G^\prime$ be the group of pairs $(\phi,g)$ satisfying…

Let $X_1$ and $X_2$ be the two components arising when a puzzle variety breaks in the geometric Littlewood–Richardson degeneration, and let $X_3$ be the puzzle variety correspondin…

Twisted Bost conjecture. The map $\mu_\sigma^{\mathcal A}$ is an isomorphism for $j=0,1$. The conjecture is presented as an unconditional analogue of the twisted Baum–Connes conjec…

Let $\Gamma$ be a torsion-free discrete group. The canonical trace on the reduced group $C^*$-algebra of $\Gamma$ induces a map on its $K$-theory, whose range is a subgroup of…