34 problems

Tsygan's formality conjecture for chains. Kontsevich's deformation quantization gives an $L_\infty$-quasiisomorphism of $L_\infty$-modules between

Let $M$ be a manifold with first Chern class $c_1(M)=0$, and let $D$ be the algebra of polydifferential operators on $M$. A BV$_\infty$-structure conjecture asserts that $D$ admits…

Kontsevich's tangent-map conjecture. The induced map

Let $Y\to X$ be a symplectic embedding of a nonformal simply connected manifold $Y$ into a simply connected compact manifold $X$. Blow-up conjecture. The blow-up of $X$ along $Y$ i…

Let $M$ be a simply connected compact symplectic manifold. The Hard Lefschetz condition is the standard Lefschetz isomorphism condition associated with the symplectic form, and a c…

Let $A_n$ be the free associative algebra on generators $x_1,\dots,x_n$, regarded as a Lie algebra with bracket $[a,b]=a\cdot b-b\cdot a$. Let $C^{\bullet}(A_n,A_n)$ be its Lie coc…

Relative formality conjecture. The DGLA, and more generally the $G_{\infty}$-algebra, of polydifferential operators compatible with $I$ is formal.

Let $cmathfrak h$ be an abelian subalgebra of a Lie algebra $cmathfrak g$, let $cmathfrak m$ be a complementary subspace, and write…

Let $K$ be a finitely generated free ${\cal D}_X$-module, and let $SK$ denote the associated chiral algebra. Let $\mathrm{def}_{SK}$ be its deformation pro-$*$-Lie algebra, and let…

Compatibility conjecture. The diagram of curved $L_\infty$-morphisms

Let $G$ be a connected graph, and let ${\mathscr A}_G$ denote its connected subgraph arrangement. For an arrangement ${\mathscr A}$, $k$-formality is the hierarchy of formality pro…

For $n\in{\mathbb N}$, let ${\mathscr H}_n$ denote the hyperpolygonal arrangement. An arrangement is $k$-formal when it has the stronger formality property indexed by $k$, with…

Mandell's conjecture. Every $n$-fold suspension is $\mathcal{E}_n$-formal.

Let $\mathcal{K}$ be a flag simplicial complex, and consider the grading on $H^*(\mathcal{Z}_\mathcal{K};\mathbb{Q})$ specified in the source's preceding theorem. Koszul formality…

Conjecture on negative quaternionic–Kähler manifolds. There are no closed simply-connected negative quaternionic–Kähler manifolds other than the known symmetric examples.

Let a Lie-quasi bialgebroid be a quasi-bialgebroid generalising Lie bialgebroids and quasi-bialgebras, and let a quasi-quantum groupoid be a topological deformation of the standard…

A manifold of special holonomy is a Riemannian manifold whose holonomy group is a special holonomy group. A space is formal when its rational homotopy type is determined by its coh…

Let $M$ be a formal manifold with $\HDR^1(M)=0$. Let the IBL-infinity chain model be the twisted $\IBLInfty$-algebra on cyclic cochains of de Rham cohomology, and let the canonical…

Let $M$ be a connected, oriented, closed Riemannian manifold with $\HDR^1(M)=0$, and suppose that $M$ is formal. In the situation of the equivalence conjecture, let $\PMC$ be the C…

Let $V$ be a differential Poincaré duality algebra with \H^0(V)=\operatorname{Span}\{1\} and \H^1(V)=0, and suppose that $V$ is formal as a differential graded algebra. A Poinc…

Let $\mathsf{fGC}_2$ be the full graph complex in dimension $2$, and let $H^1(\mathsf{fGC}_2)$ denote its first cohomology. Kontsevich's vanishing conjecture. … This vanishing woul…

A simply-connected compact symplectic manifold is a compact symplectic manifold with trivial fundamental group, and the hard Lefschetz property is the usual Lefschetz isomorphism p…

Formality-choice independence conjecture. The bijection between $\mathsf{TL}$ and the set of formal series in … .

Formality-independent bijection conjecture. There is a construction of a bijection between $\mathsf{TL}$ and the set of formal series in $$ which bypasses the use of Kontsevich's f…

Shoikhet's conjecture. Such an $L_\infty$-morphism should also be compatible with the cup product.