Representations of $p$-adic Groups and the Local Langlands Correspondence

The local Langlands correspondence predicts a natural bijection between irreducible smooth representations of $\text{GL}_n(F)$ (where $F$ is a $p$-adic field) and $n$-dimensional Weil-Deligne representations of the Weil group $W_F$. We review the proof for $\text{GL}_2$ via the theory of types, then discuss the revolutionary approach of Fargues and Scholze, who construct a geometrization of the correspondence using the Fargues-Fontaine curve and the stack $\text{Bun}_G$ of $G$-bundles, placing the local Langlands program \in the framework of geometric Langlands.