12 problems

Let $X$ and $X^!$ be symplectic dual cones with Poisson brackets of degree two. Let $\operatorname{HP}_0(\mathcal{O}(X))$ denote the zeroth Poisson homology of the coordinate ring…

Let $\mathfrak{g}$ be a semisimple Lie algebra with Weyl group $W$, and let $X$ be the corresponding nilpotent cone. For each irreducible representation $\chi$ of $W$, let…

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 $\mathcal M_H$ be the Higgs branch, viewed as a quiver variety, and let $\widetilde{\mathcal M}_H\to\mathcal M_H$ be its symplectic resolution. Write $H\\!P_0(\mathcal M_H)$ fo…

Let $\mathfrak{N}$ be an affine cone over $\mathbb{C}$ admitting a projective symplectic resolution $\mathfrak{M}$, and let $\mathscr{N}$ be the quasi-universal deformation of…

Let $V$ be a symplectic vector space and let $G<\mathsf{Sp}(V)$ be finite, assuming that $V/G$ admits a symplectic resolution. Poisson and Hochschild homology conjecture. The canon…

Let $X$ be an irreducible affine Poisson variety admitting a symplectic resolution $\widetilde{X}\to X$. Let $A_\hbar$ be a deformation quantization of $\mathcal{O}_X$, and let…

Alev's wreath-product conjecture. The equality

Degree bound conjecture. This equation has no solution of degree strictly higher than $4(n-1)$.

Alev's completeness conjecture. For every integer $n\geq 2$, these $\pi(n)$ constructed graphs are the only solutions of the equation.

Simple-graph uniqueness conjecture. For every integer $n\geq 2$, there exists a unique polynomial of degree $4(n-1)$ in $S^W_{\mathfrak{sl}_2}$ solving the equation. It is represen…

Alev's conjecture.