21 problems

Loday's proposal. These data should satisfy

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\mathbf{\Gamma}^{[a,b]}(\underline{w}_0)$ and…

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\widehat{\mathbf{\Gamma}}^{[a,b]}(\underline{w}_0)$ be the c…

Let $(S,n,{\bf a})$ be a parametrized surface and let $R(S,n,{\bf a};k)$ be the dga generated by diagrams of $k$ strands between the corresponding collections of arcs. Let $\mu_m$…

Benson's generation conjecture. The category $\mathcal{D}^b(C^*BG)$ is generated as a thick subcategory of $\mathcal{D}(C^*BG)$ by the objects $\operatorname{Hom}_{C_*G}(k,X)$ for…

Let $k$ be the base field, let $M\in M_3(k)$, and let $\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ denote the associated cochain DG skew polynomial algebra. Write…

Infinite free-extension naive lifting conjecture. If

Let $A$ be a connective dga and let $S$ be a one-dimensional $A$-module. Let $E\coloneqq {\mathrm{\normalfont\mathbb{R}}}\mathrm{End}_A(S)$ be the derived endomorphism dga of $S$.…

Let $T_b^{\lambda,\underline{N}}$ be the dg-enhanced KLRW algebra and let $({}_{dg}Q_b^{{\underline{N}}},d_N)$ be the dg-quiver Schur algebra, with the two maps of graded dg-algebr…

Let $({}_{dg}Q_b^{{\underline{N}}},d_N)$ be the dg-quiver Schur algebra defined as the dg-endomorphism algebra of the direct sum of the dg-enhanced modules…

Let $G_{\rho}^{{\underline{N}}}$ and $Y_{\rho}^{{\underline{N}}}$ be the dg-enhanced and ordinary cyclic modules constructed from the tensor-product data, with differential $d_N$ o…

Let $R$ be a Gorenstein singularity, let $M$ be a maximal Cohen–Macaulay $R$-module, and set $A:=\operatorname{End}_R(R\oplus M)$. Let $e\in A$ be the idempotent corresponding to…

Let $\mathscr R$ be a smooth finite-dimensional DG algebra, and let $\mathscr{P}\!\mathit{erf}\!\operatorname{--}\mathscr R$ denote its DG category of perfect modules. A phantom is…

Let $Z$ be a projective scheme, let ${\Bbbk}$ be the base field, and let $\mathscr R$ be a finitely-dimensional DG algebra. Suppose that the DG category…

Formality conjecture. If $Q$ is connected and not Dynkin, then

Let $X$ be a projective K3 surface with a generic polarization $H$, and let $F$ be an $H$-polystable coherent sheaf on $X$. Kaledin–Lehn's formality conjecture. The differential gr…

Let $R$ be the DG-algebra under consideration, and let ${\mathcal F}$ be the full subcategory of ${\mathsf{D}}(R)$ consisting of the relevant DG-flat objects. Let…

Calabi–Yau contraction isomorphism conjecture. Under the formality assumption, the induced morphism of dg $k$-modules

Let $\Gamma$ be any tree, let $\mathscr{B}_\Gamma$ be the DG-algebra arising from the plumbing construction, and let $\mathscr{G}_\Gamma$ be the corresponding Ginzburg DG-algebra.…

Let $S$ be the underlying scheme, and let $\mathcal{B}$ be a sheaf of $\mathbb{Z}$-graded differential-graded $\mathcal{O}_S$-algebras and $\mathcal{A}$ a sheaf of $\mathbb{Z}$-gra…

Noncommutative distinguished-triangle conjecture. There is a natural distinguished triangle