25 problems

Tsygan's generalized Gerstenhaber formality conjecture. For any associative algebra $A$, the Hochschild chain complex $C_{\bullet}(A,A)$ is a $G_{\infty}$ module over the…

Let $X$ be a compact smooth space. The modified Hochschild-Kostant-Rosenberg maps $I^K$ and $I_K$ are the maps obtained by twisting the Hochschild-Kostant-Rosenberg isomorphism by…

Let $(X,D)$ and $(X',D')$ be log smooth pairs, and let $E$ be a strong Fourier–Mukai kernel on $X\times^{\log}X'$. Let $\pi:X\times^{\log}X'\to X\times X'$ be the projection and re…

Let $(X,D)$ be a log smooth pair, and let $E_1\to E_2\to E_3\to E_1[1]$ be a distinguished triangle in $D(X\times^{\log}X)$ whose terms are strong log Fourier–Mukai kernels. Additi…

Let $B$ be a finite-dimensional algebra over an algebraically closed field $k$. Write $\operatorname{HH}_n(B)$ for the Hochschild homology groups of $B$. The algebra $B$ is (homolo…

Decategorification isomorphism conjecture. The injective algebra homomorphism above is always an isomorphism.

Let $R$ be a commutative ring, and let … be the map classifying the filtered nodal curve over $R$. Denote the corresponding filtered nodal curve by $\mathcal{N}_{\mathrm{fil}}$, an…

Let $C:\operatorname{Spec} R\to \mathcal{M}_{cub}$ be a cubic curve, and let $\widehat{C}$ denote its formal completion, a formal group over $R$. Write…

Let $\Theta$ be a tangential structure, let $R$ be its associated ring spectrum, and suppose $\operatorname{Flow}^{\Theta} \simeq \operatorname{Perf}({}_R\operatorname{mod}_R)$. Fo…

Let $\mathcal{T}$ be a smooth proper dg category, and let $\operatorname{Sym}^n\mathcal{T}$ denote its $n$th symmetric power in the sense of Ganter–Kapranov. Write…

Let $K\subset S^3$ be a framed oriented knot and let $X\in\operatorname{Ch}^-(\mathsf{BN}_n)$ be a complex of $n$-strand Bar–Natan 1-morphisms. Let $C(K,V_n)$ denote the $V_n$-colo…

Let $(X,\mathcal{C}^\infty)$ be a homologically regular reduced differentiable space. Let $\mathscr{E}_\bullet(\mathcal{C}^\infty_X)$ be its diagonal sheaf complex and let…

Let $M$ be a closed oriented C^-manifold, let $\Omega(M)$ denote its de Rham algebra, and let $\Omega(M)^\vee[-\dim M]$ be the shifted dual complex. Write…

Indecomposability criterion. The category $\mathcal{B}$ is indecomposable if and only if

Let $(\mathscr{V},\otimes,I)$ be a presentably symmetric monoidal $(\infty,1)$-category. Let $\mathscr{M}\mathrm{od}_{\mathscr{V}}$ be the $(\infty,2)$-category whose objects are…

Let $\formalgroup$ be a formal group over a perfect field $k$. For a simplicial commutative $k$-algebra $A$, let $\theta(A)$ be its underlying $E_\infty$-algebra, and write…

Let $\mathfrak g$ be the Lie algebra under consideration, let $s$ denote the degree-sign involution, and let ${{\mathbf A}}_\lambda$ be the corresponding filtered quantization para…

Let $Q$ be the polynomial ring and $f$ a polynomial as in the source, let $n$ be the relevant degree, and let $c_f\in\mathbb{C}$ be the constant such that the canonical pairing on…

Let $G$ be a $p$-adic group, let ${\rm Sm}$ be the category of smooth modules and ${\rm Adm}\subset {\rm Sm}$ the subcategory of admissible modules. Let $HH_0^c({\rm Sm})$ and…

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero, and let $\mathcal{A}\subset\mathcal{D}^b(X)$ be an admissible subcategory, meaning that its inclusio…

Let $\sigma \in {\mathfrak{B}}_{n}$, and let $m(\widehat{\sigma}) \subset A \times I$ be the mirror of its annular closure. The notation $A^{k,n-k}$ denotes the Chen–Khovanov–Strop…

Let $X$ be an irreducible affine Poisson variety admitting a symplectic resolution $\widetilde{X}\to X$. Let $A_\hbar$ be a deformation quantization of $\mathcal{O}_X$, and let…

Let $T$ be a saturated dg-category. Consider its algebraic classes in Hochschild homology, namely classes of the form $\alpha_E(\operatorname{Id}_E)$ for objects $E\in T$, and call…

Let $X$ be a smooth projective variety, let $\D^b(X)$ denote its bounded derived category of coherent sheaves, and let $\CA\subset \D^b(X)$ be an admissible subcategory. Write…

Hesselholt–Rains conjecture. For every prime $p\geq 2$, the classes $r^{(p^\ell)}$ are nonzero and generate the $p$-torsion of $\Pi_{\mathrm{cyc}}$. Moreover, $\Pi_{\mathrm{cyc}}$…