Stabilization Properties of Chains of Symmetric Ideals

In the non-Noetherian ring $k[x_1,x_2,\ldots]$, ideals $I$ which are invariant under an action of the infinite symmetric group enjoy certain finiteness properties. In this expository talk, we will discuss some advances in the study of these ideals and their associated truncations $I\cap k[x_1,\ldots,x_n]$. In particular, we will discuss algebraic properties of the truncations as $n$ tends to $\infty$. This talk will include plenty of examples and a mini introduction to free resolutions and Betti tables.