Given a H\"older family of functions $F$ and a finitely irreducible CGDMS $\Phi$ encoded by a symbolic representation $E_A^{\infty}$, one may associate to each coding $\omega\in E_A^{\infty}$ a Birkhoff average $\xi(\omega)$ called the $F$-exponent of $\omega$, should it exist. The ergodic optimization of these exponents by way of zero-temperature style limits is important to the characterization of Birkhoff spectra for CGDMSs. In this talk, we will introduce the objects at play in this optimization problem and the results which address this problem for a large collection of possible families $F$.

Let $W$ be the Witt algebra of vector fields on the punctured complex plane, and let $\text{Vir}$ be the Virasoro algebra, the unique nontrivial central extension of $W$. $\text{Vir}$ is an infinite-dimensional Lie algebra that is ubiquitous in representation theory and important in physics. The symmetric algebras of $\text{Vir}$ and $W$ are polynomial rings in infinitely many variables which also have a Poisson structure induced from the bracket on the Lie algebras. Poisson ideals of $\text{Sym(Vir)}$ and $\text{Sym(W)}$ approximate two-sided ideals in the corresponding universal enveloping algebras and give evidence for the two-sided structure of the enveloping algebras.

We study prime Poisson ideals in these symmetric algebras, focussing on understanding Poisson primitive ideals, which are the Poisson cores of maximal ideals of $\text{Sym(Vir)}$ and of $\text{Sym(W)}$. Surprisingly, these give a way to do finite-dimensional algebraic geometry with these infinite polynomial rings! We give a complete classification of maximal ideals of $\text{Sym(W)}$ which have nontrivial Poisson cores. We then lift this classification to $\text{Sym(Vir)}$, and use it to show that if $q$ is a nonzero complex number, then $(z-q)$ is a maximal Poisson ideal in $\text{Sym(Vir)}$.

This is joint work with Alexey Petukhov.