88 problems

Conjectured coefficient formula. The coefficients satisfy

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 $\mathcal{X}$ 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 $A$ be the algebra of regular functions on a quasihomogeneous isolated surface singularity $X$ defined by $Q(x,y,z)=0$ over a field of characteristic $p$, with the notation…

Let $A=C^\infty(X)$ be the algebra of smooth functions on a smooth real manifold $X$, and let $C^\bullet=C^\bullet(A,A)[1]$ be the suspended local Hochschild complex, whose cohomol…

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 $Y$ 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…

For each $n,N\geq 1$, let ${\underline{\operatorname{Aut}}}^{\leq N}(A_{n,\mathbb Q})$ and ${\underline{\operatorname{Aut}}}^{\leq N}(P_{n,\mathbb Q})$ denote the filtered automorp…

Poisson-cohomology conjecture. There is an isomorphism of vector spaces

Poisson linearization conjecture. If the restricted Lie algebroid $T^*_S M$ integrates to a compact, source 1-connected Lie algebroid, then $pi$ is linearizable around $S$.

Relative Poisson cohomology conjecture. Let $pi$ be a Poisson structure with a compact symplectic leaf $S\subset M$. If

Birational torus conjecture. There is a birational equivalence between the symplectic variety $({\mathcal O}_1\times{\mathcal O}_2\times\bar{{\mathcal O}}_3)///G$ and a complex tor…

Birational torus conjecture. There is a birational correspondence between the variety $({\mathcal O}\times {\mathcal O}_-)///B_+$ described above and a symplectic torus of the same…

Let $\mathfrak{g}$ be a simple Lie algebra. The polynomial algebra on $\mathfrak{g}^*$ is $S\mathfrak{g}$. An invariant Poisson bracket is a Poisson bracket on $S\mathfrak{g}$ inva…

Poisson vector bundle conjecture. (1) Every homogeneous Hermitian Poisson vector bundle over ${\mathfrak g}^*$ is Poisson-trivial. (2) There is a one-to-one correspondence between…

Let $\mathfrak{bms}_3^*$ be the dual of the centrally extended $\mathfrak{bms}_3$ algebra, let $c_1$ be its first central charge, and let $\{,\}_\mathrm{LP}^{\mathfrak{bms}_3^*}$ a…

Polarized semiclassical Yau–Tian–Donaldson conjecture. The following are equivalent:

Semiclassical Yau–Tian–Donaldson conjecture. The triple $(X,\sigma,\alpha)$ is Poisson K-polystable if and only if there exists $\epsilon>0$ such that, for every…

Bondal's conjecture. For every $k\in\{0,1,\dots,r-1\}$, $D_{2k}(\pi)$ has an irreducible component of dimension at least $2k+1$.

Cyclicity conjecture. The Hamiltonian $h^{(N)}$ is the cyclic variable of the quadratic Poisson structure.

Let $K_0\backslash\mathcal{G}_0$ be the quotient under consideration, let $\lambda$ be a dominant coweight, let $\omega_i$ be the fundamental coweights for $1\leq i\leq n-1$, and s…

Let $W$ be a vertex algebra that is a finite extension of a vertex algebra $L$, and let $X_W$ and $X_L$ denote their associated Poisson varieties. Dominance conjecture. The morphis…

Arbitrary-genus symplectic-leaf conjecture. For every integer $k\in\mathbb{N}$, an appropriate choice of $U$ produces a sample of symplectic leaves that are topologically Riemann s…