turkmath.org

Given a sequence $\lbrace G_{n}\rbrace$ of groups and vector spaces $\lbrace V_{n}\rbrace$ where each $G_{n}$ acts on $V_{n}$, representation stability studies the eventual/uniform behavior of $V_{n}$ as a $G_{n}$-representation, as $n$ varies. One of the most robust and applicable variants is when $G_{n}$ is the symmetric group on $n$ letters, categorified by the framework of FI-modules. In addition to giving an overview of FI-modules and consequences of their finite generation, I will discuss
applications such as the cohomology of configuration spaces and homology of congruence subgroups, where several stabilization results can be obtained even when explicit computations are lacking.