Let W be the Witt algebra of polynomial vector fields on the punctured complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central extension of W. We discuss work in progress with Alexey Petukhov to analyse Poisson ideals of the symmetric algebra of Vir. We focus on understanding maximal Poisson ideals, which can be given as the Poisson cores of maximal ideals of Sym(Vir) and of Sym(W). We give a complete classification of maximal ideals of Sym(W) which have nontrivial Poisson cores. We then lift this classification to Sym(Vir), and use it to show that if $\lambda \neq 0$, then $(z- \lambda)$ is a maximal Poisson ideal of Sym(Vir).
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