26 problems

Duflo formula for Q-manifolds. 1. The map

Kontsevich's tangent-map conjecture. The induced map

Let $H(\underline{t};q)$ be the rank-one double affine Hecke algebra with parameters $q$ and $\underline{t}$. Here $HH^i$ denotes its $i$-th Hochschild cohomology group, and…

Let $M$ be an exact symplectic manifold satisfying Assumption, and let … be the natural open–closed string map from symplectic cohomology to the Hochschild cohomology of the Fukaya…

Let ${\cal C}$ be an $A_{\infty}$-category such that $HH^1({\cal C})$ is abelian and the connected automorphism group $G_{\cal C}=Aut^0({\cal C})$ is a finite-dimensional commutati…

Let $G$ be a finite group, let $k$ be a field whose characteristic divides the order of $G$, and let $B$ be a block of the group algebra $kG$ with defect group $P$. Linckelmann's c…

Belmans–Smirnov's conjecture. If $X$ is neither (co)minuscule nor (co)adjoint, then $X$ is not Hochschild global; equivalently, for some $i>0$ and some $j$,

Let $V$ be a symmetric monoidal category and let $\mathcal{P}$ be a $V$-enriched $2$-operad. Let $\mathcal{P}_1$ be the $1$-operad obtained by restricting $\mathcal{P}$ to the $1$-…

Open-closed Deligne conjecture. For any open-closed homotopy algebra $(\mathfrak l,\mathfrak q)$, there is a $G_\infty$ algebra structure on the space…

Let $X$ be a log scheme for which the constructions in the paper are defined, let $i:X\to B$ be the local complete intersection morphism used to define logarithmic Hochschild cohom…

Let ${\mathrm{K}}_m^n$ be the extended Khovanov arc algebra indexed by $m$ and $n$, and let $\operatorname{HH}^2_{i-2}({\mathrm{K}}_m^n,${mathrm{K}}m^n)…

Non-triviality conjecture. The cyclic cohomology class $[\mathfrak{c}_\omega]$ and the associated Hochschild cohomology class $[\mathfrak{h}_\omega]$ are non-trivial. These classes…

Deligne's conjecture. There exists a natural action of the operad $\mathsf{Chains}(C_2)$ on the Hochschild complex $\mathcal{C}^\bullet(\mathcal{A},\mathcal{A})$ for any associativ…

Let $\operatorname{\textit{Quilt}}$ be the operad of quilts, and let $H_k(\operatorname{Quilt})$ denote its homology in degree $k$. Acyclicity conjecture for the Quilt operad.…

Let $\mathscr{X}$ be a projective symplectic quotient orbifold, and write $H\! H^\bullet(\mathscr{X})$ for its Hochschild cohomology and $H^\bullet_\mathrm{orb}(\mathscr{X})$ for i…

Let $\mathscr{X}$ be a symplectic quotient orbifold and let $\mathscr{A}$ be an appropriate quantization of $\mathscr{X}$. Ginzburg–Kaledin conjecture. There is an algebra isomorph…

Let $A$ be an algebra, let $C_{sg}(A,A)$ denote its singular Hochschild cochain algebra, and let $\operatorname{\mathsf{sg}}_{dg}(A)$ be the dg singularity category of $A$. Write…

Let $A$ be a finite ordered set, let $\Bbbk$ be a commutative unital ring, and let $\mathbf{Pl}_A$ denote the plactic monoid on $A$. Theorem gives an injective algebra map from the…

Let $G$ be a connected compact Lie group of dimension $d$, let $BG$ be its classifying space, and let $LBG$ denote the free loop space of $BG$. Write $S_*(G)$ for the singular chai…

Let $\mathcal{T}$ be a full admissible subcategory of the derived category of a smooth projective variety. Assume that $\mathcal{T}$ is hyper-Kähler of dimension $2r$, and write…

Free-loop-space F-manifold conjecture. These ring isomorphisms can be lifted to isomorphisms of cohomology F-manifolds. This conjecture extends the known algebraic structures relat…

Let $\mathcal A$ be an associative algebra equipped with a differential … For a theory with a single D-brane, the deformation data are encoded by $(\mathcal A,Q)$, and their equiva…

Let $X$ be a smooth algebraic symplectic variety, and let $Y,Z\subset X$ be smooth Lagrangian submanifolds. Write $H\!H^{\bullet}(\mathscr{C})$ for the Hochschild cohomology of a t…

Let $G$) be a connected compact Lie group of dimension $d$, let $LBG$ denote the free loop space of its classifying space, and let $S_*(G)$ and $S^*(G)$ denote the singular chains…

Let $M$ be a compact oriented $d$-dimensional smooth manifold, let $LM=map(S^1,M)$ be its free loop space, and let $S^*(M)$ denote the differential graded algebra of singular cocha…