388 problems

Let $X$ be a variety admitting crepant resolutions, including commutative resolutions and non-commutative crepant resolutions. Van den Bergh's conjecture. All crepant resolutions o…

Artin's conjecture. Birationally, there is a short list of noncommutative projective surfaces, with the generic case being a Sklyanin algebra.

The Hochschild-to-cyclic spectral sequence for an c4b1c4b7b3bcb1c4b1-algebra $A$ is … Here $A$ is smooth and proper over a field of characteristic $0$. Kontsevich–Soibelman conject…

Let $R$ be a Gorenstein toric algebra, meaning the coordinate ring associated to a Gorenstein cone. An NCCR of $R$ is an algebra of the form $\Lambda=\operatorname{End}_R(M)$ for…

Let $A$ be a connected regular algebra of exponential growth, where regularity means finite global dimension together with the stated Gorenstein property. Bondal's coherence conjec…

Blanc's lattice conjecture. If $\cC$ is smooth and proper over $\C$, then

Let $A$ be a unital $C^*$-algebra with a free coaction $\delta$ of a nontrivial compact quantum group $(H, \Delta)$. Equip the join $A \circledast^{\delta} H$ with the induced free…

Let $(\Lambda,[I],K)$ be the triple corresponding to a pseudo-Anosov map $\phi\in Mod~(X)$, and let $\lambda_\phi$ be its dilatation. The integral order $\Lambda$ has class number…

Consider noncommutative extensions of integrable equations and noncommutative Lax equations arising in the study of reductions of noncommutative anti-self-dual Yang--Mills equation…

Let $\mathbb{T}^n_\theta$ be the quantum torus, let $\Delta$ be its flat Laplacian, let $q>0$, set $r=2nq^{-1}$, and suppose that either $r\neq 1$ and $r'=\max(r,1)$, or $r=1<r'$.…

Let $A$ be a connected Artin–Schelter regular algebra, meaning a connected graded regular algebra of polynomial growth. Artin–Schelter's noetherianity conjecture. Every connected A…

Vanishing Einstein functional conjecture. Suitably regular two-dimensional spectral triples will have the Einstein functional identically vanishing.

A homogeneous space of a compact Lie group is a quotient equipped with the induced smooth structure, and its dimension as a differentiable manifold is its usual manifold dimension.…

Let $\mathbb{K}$ be the ground field and let $A=\mathbb{K}\langle x_1,x_2,x_3\rangle$ be the free associative algebra on generators $x_1,x_2,x_3$. Define two operations on pairs of…

Let $B$ be a quantized coordinate ring and let $\overline{B}$ be its corresponding semi-classical limit, with the induced Poisson structure. Assume that the base field is algebraic…

Let $D$ be the Dirac operator on a Riemannian manifold, and let the noncommutative residue assign a residue to suitable powers of $D$. Connes's conjecture. The noncommutative resid…

Let $n$-soliton solutions be the multi-soliton solutions constructed on noncommutative Euclidean spaces by noncommutative Darboux and Bäcklund transformations. An isolated localize…

Let $X$ be a smooth proper complex variety, and let $\mathcal{C} \subset \operatorname{D}_{\mathrm{perf}}(X)$ be a $\mathbf{C}$-linear admissible subcategory. Write…

Let $A$ be a connected graded domain over $\bbk$, and write $\operatorname{GKdim}(A)$ for its Gelfand–Kirillov dimension. Artin–Stafford's gap conjecture. There is no connected gra…

Born's reciprocity principle. The basic underlying physical space should be the eight-dimensional phase space with coordinates $\\{x_\mu,p_\mu\\}$, and the Poincaré line element sh…

Noncommutative Borsuk–Ulam conjecture. There does not exist a $\mathbb{G}$-equivariant $*$-homomorphism

Constant-term spectral-action conjecture. The constant term of the spectral action of $\mathcal{D}_A$ on the noncommutative $n$-torus is proportional to the constant term of the sp…

Let $R$ be a Gorenstein $coperatorname{\mathbb{C}}$-algebra. A crepant resolution of $R$ is a crepant resolution of the corresponding singularity, and an NCCR of $R$ is a non-commu…

Let $E$ be a finite extension of $\mathbb Q$ with Galois group $Gal~(E|\mathbb Q)$, and let $\sigma_{n+1}:Gal~(E|\mathbb Q)\to GL_{n+1}(\mathbb C)$ be an irreducible representation…

Let $\Gamma$ be a Fuchsian group with fundamental class $[\Gamma]$, let $\sigma$ be a multiplier on $\Gamma$ satisfying $\delta(\sigma)=0$, and define … Here a…