220 problems

Bondal's conjecture. The locus $\mathsf{D}_{2k}(\sigma)$ has an irreducible component of dimension at least $2k+1$ for every such integer $k$. The conjecture predicts that degenera…

Let $A_n(\mathbb{C})$ be the $n$-th Weyl algebra, and let $\mathbb{C}^{2n}$ carry its standard Poisson structure. A polynomial symplectomorphism of $\mathbb{C}^{2n}$ is a polynomia…

Let $G$ be a connected, simply connected, complex simple algebraic group, and let a Poisson bracket on $G$ belong to a Belavin–Drinfeld class. Gekhtman–Shapiro–Vainshtein conjectur…

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…

Flat Darboux coordinate conjecture. These quantities are flat Darboux coordinates for the reduced Poisson structure. The source states that the associated result was explicitly pro…

Let $\mathfrak{g}$ be a Lie algebra, and let $\left\{\,, \,\right\}$ denote its Lie–Poisson bracket. A family of functions on $\mathfrak{g}^*$ is complete in the usual integrabilit…

Let $\mathbb{K}$ be the ground field and let $A=\mathbb{K}\langle x_1,x_2,x_3\rangle$ be the free associative algebra on generators $x_1,x_2,x_3$. Define two operations on pairs of…

Let $B$ be a quantized coordinate ring and let $\overline{B}$ be its corresponding semi-classical limit, with the induced Poisson structure. Assume that the base field is algebraic…

Let $\mathcal{X}$ be a stacky curve, let $\alpha$ be a fixed parabolic structure, and let $\widetilde{\mathcal{M}}^{ss}_H(\mathcal{X},\alpha)$ denote the moduli stack of semistable…

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…

GGT's normalizer conjecture. Every integrable irreducible representation of ${\cal B}$ can be extended to a quantization of $({\cal N}({\cal B}),{\cal B})$, and no nontrivial quant…

GGT's basic-set conjecture. For every basic set ${\cal B}\subset {\cal P}(M)$ whose generated Poisson algebra is finite-dimensional, the associated quantization should satisfy the…

Let $G$ and $G'$ be connected, simply connected semi-simple algebraic groups, and let $\psi:G'\times G'\to G$ be an admissible embedding. Let $\tau$ be the triple obtained from the…

Let $\operatorname{Diff}(S^1)$ denote the group of diffeomorphisms of the circle $S^1$, equipped with its group structure. A Poisson-Lie structure is a Poisson structure for which…

Poisson convexity conjecture. The image

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…

In the setting of the Casimir-family criterion, let $M$ carry compatible Poisson structures $\{,\}_{1}$ and $\{,\}_{2}$, let $F_{l,\lambda}$ and $W_1(m)$ satisfy the preceding hypo…

Let $M$ be a manifold with compatible Poisson structures $\{,\}_{1}$ and $\{,\}_{2}$. Let $L$ be a finite set with $r$ elements, and let $F_{l,\lambda}$, for $l\in L$ and…

Let $M_{1}$ and $M_{2}$ be manifolds equipped with bihamiltonian structures, and equip $M_{1}\times M_{2}$ with their product bihamiltonian structure. Product-flatness conjecture.…

Let a homogeneous bihamiltonian structure have type $\left(2k_{1}-1,2k_{2}-1,\dots,2k_{l}-1\right)$. Tensorial criterion for Kronecker structures. There exist $N>0$ and natural ten…

Let $A$ be an associative algebra with generators $x_1,\ldots,x_n$ and defining relations … where $p_{ij}\in\mathbb{C}[x_1,\ldots,x_{i-1}]$ if $i>j$, $(h_{ij})$ is an antisymmetric…

Let $P_1$ and $P_2$ be Morita equivalent Poisson manifolds, with dimensions $m_1=\dim P_1$ and $m_2=\dim P_2$. The preceding result gives an isomorphism … when $P_1$ is unimodular.…

Let $\mathsf{H}(V)$ be the Frobenius manifold associated with a smooth projective variety $V$, and assume that it is semisimple. Let $\{\!\{u_a(x),u_b(y)\}\!\}_0$ and…

Let $Y$ be a symplectic singularity, let $y\in Y$ be a closed point, and let $Y_{y,0}$ be the transversal symplectic slice in the formal product decomposition … A $\mathbb{G}_m$-ac…