110 problems

Kashiwara–Vergne conjecture. The specified set of Kashiwara–Vergne equations has a solution in the group of tangential automorphisms of $\widehat{\mathfrak{f}}_2$.

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

Landsman quantization reconstruction conjecture. The classical limit functor is almost inverse to the quantization functor defined by Landsman, in the sense that the classical limi…

Let $W_{n,\mathbb{C}}$ be the $n$-th Weyl algebra over $\mathbb{C}$, and let $P_{n,\mathbb{C}}$ be the $n$-th commutative Poisson algebra. Kontsevich's conjecture. The automorphism…

Let $A$ be the algebra of functions on an affine space, let $V_A$ be the graded Lie algebra of polyvector fields on that affine space, and let $C^{\bullet}(A)$ be the differential…

Quantizability conjecture. Any Lie bialgebroid is quantizable.

A deformation quantization is a quantized deformation of an algebraic or geometric structure, and the motivic Galois group is the symmetry group arising from the theory of motives.…

Let $A_{n,\mathbb{C}}$ be the Weyl algebra and let $P_{n,\mathbb{C}}$ be the commutative polynomial algebra equipped with its Poisson bracket. For an infinite prime $[p]$, let … be…

Let $L$ be an $ffrak{h}$-base manifold, let $P$ be an $ffrak{h}$-manifold, and let $pi$ be a Poisson dynamical bracket on $P$ over $L$. A quantization of the Poisson dynamical…

Kanel-Belov–Kontsevich conjecture. The possibly larger group ${\mathrm{Aut}}({\mathbb C}^{2n})$ is the automorphism group of the Weyl algebra ${\mathscr A}$.

Let $n\geq 0$, let $X$ be a nice enough derived algebraic stack, and let $\widehat{HH}^{E_{n+1}}(X)$ denote its formal higher Hochschild cohomology, so that…

Let $V$ be a finite-dimensional vector space, let $x_i$ be coordinates corresponding to a basis of $V^*$, and let $R_{ij}(\bbar{hbar})$ be the formal series defining the PBW algebr…

Let $F$ be a quadratic magnetic tensor, so that the magnetic curvature is constant. Let $f$ and $g$ be functions on the phase space, let $\Psi(x)$ denote the symplectic tensor at…

Supersymmetric-action conjecture. The supersymmetric action can be obtained by an appropriate deformation $\bmod\,\hbar$ of the Kähler metric.

Cycle-removal conjecture. The Weyl star-product associated with ${\ast^K_\hbar}$ contains no cycles and is obtained from ${\ast^K_\hbar}$ by ignoring the graphs with cycles.

Let $M$ be a smooth manifold with a Poisson structure $\pi_0$, and let $\widehat{A}(\pi_0)\in H^{\mathrm{ev}}(M,\mathbb{C}[[t]])$ be the class obtained from the periodic cyclic for…

$A_\infty$-Formality conjecture. The associative differential graded algebra $D^\bullet_\mathrm{poly}(\mathbb R^d)$ is quasi-isomorphic, as an associative algebra, to its cohomolog…

Fedosov curvature-exponential conjecture. The part of the Fedosov star-product expression that is purely in $\alpha$ contains

Let $(\mathfrak{g},\sigma)$ be a symmetric pair with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, and let $E(X,Y)$ be the function defined from graphs for…

Let $M$ be a smooth manifold with a connection, and let the formula defining the projectively equivariant quantization be used to define a deformation of the algebra of symbols on…

Let $\pi_n^{(\lambda)}$ denote the deformed quasiprobability associated with energy level $n$, parametrized by $\lambda$, and let $H>0$ be an energy value. Negativity conjecture. F…

A natural star product is a formal associative deformation of the pointwise product whose bidifferential operator at order $r$ differentiates each argument at most $r$ times. The g…

Let $R$ be a finitely generated smooth commutative algebra over $\mathbb Z$, let $P_{n,R}$ be the polynomial Poisson algebra, and let $A^{\hbar}_{n,R[\hbar]}$ be the algebra over…

Let ${\mathfrak{X}}$ be a complex symplectic manifold, and let $\mathcal{L}_i$ for $i=0,1$ be objects of $D^b_{\rm rh}(\mathcal{W}_{\mathfrak{X}})$ supported by Lagrangian submanif…

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