154 problems

For every smooth and proper differential $\mathbb{Z}/2$-graded category $\mathcal{C}$ over $\mathbb{C}$, the Getzler–Gauss–Manin connection $\nabla^{\mathcal{C}}_{\partial_t}$ in t…

For every smooth proper differential graded category $\mathcal{A}$, the rational Chern character is surjective onto the rational noncommutative Hodge classes:…

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…

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 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…

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 $X$ be a normal algebraic variety with Gorenstein singularities, and let … be two crepant resolutions by schemes or Deligne–Mumford stacks. An equivalence of triangulated categ…

Let $R$ be an equidimensional normal Gorenstein ring. A noncommutative crepant resolution of $R$ is an endomorphism ring of a modifying $R$-module with finite global dimension, whi…

Let $\mathcal{T}$ be a smooth proper dg-category over $\mathbb{C}$. Write $K_{\operatorname{top}}(\mathcal{T})$ for its topological K-theory spectrum,…

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

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 $A$ be a noetherian, connected graded domain over ${\Bbbk}$ with $\operatorname{GKdim} A=3$. Its graded quotient ring has the form $Q_{gr}(A)\cong D[t,t^{-1};\alpha]$, where…

Perry's noncommutative integral Hodge conjecture. The homomorphism

Quasi-bisymplecticity conjecture. The triple $({\mathcal{B}}(\Gamma_n),\Phi,\omega_n)$ is a quasi-bisymplectic algebra. Furthermore, the double quasi-Poisson bracket…

Let Equation … has a unique solution with a given appropriate boundary condition at infinity. This is an informal conjecture attributed to Nekrasov; the source does not specify the…

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…

Multiplicity inequality. The following inequality holds:

Let $M$ be a compact complex manifold, let $obreak\lambda$ be a line bundle on $M$, and let $\boldsymbol1_\lambda$ denote its identity differential operator. Write…

Let $C$ be a genus-$g$ Heegaard surface with fundamental group $\pi_1(C)$, let $X$ be the associated three-manifold, and let $L_+$ be one of the two lattices in the Heegaard splitt…

Let $R$ be the base algebra, let $A$ be a friendly Calabi–Yau $R$-algebra of dimension $3$, let $F$ be a smooth $R$-algebra, and let…

Let $(D,\omega,\xi)$ be the symplectic data used in the universal construction, and let $A$ be a friendly algebra satisfying the stated Calabi–Yau condition. Theorem main(ii) repre…

Let $\mathfrak{a}$ be a simple Lie algebra, let ${\mathbf d}\geq 1$, and let $\psi_{\mathbf d}$ be the real-valued polynomial obtained by restricting the potential $\Psi$ to the sp…

Kontsevich–Soibelman conjecture. For any saturated DG algebra $A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ over a field $K$ of characteristic $0$, the Hodge-to-de Rham spectral seq…