Explore the intricacies of the mod $p$ Langlands correspondence in this one-hour lecture by Florian Herzig at BIMSA. Delve into an analog of the classical local Langlands correspondence, focusing on the relationship between Galois representations and representations of $p$-adic groups over characteristic $p$ coefficient fields. Examine the well-established case for $\mathrm{GL}_2(\mathbb Q_p)$ and investigate the challenges in extending this correspondence to other reductive groups, particularly $\mathrm{GL}_2(K)$ where $K$ is a nontrivial finite extension of $\mathbb Q_p$. Survey the current conjectures and recent advancements in the mod $p$ Langlands correspondence for $\mathrm{GL}_2(K)$, gaining insights into this open problem in modern number theory and representation theory.