Broadly speaking, for G a connected reductive group over a local field F, the Langlands program is the endeavor of relating Galois representations (more precisely, "L-parameters"---certain homomorphisms from the Weil-Deligne group of F to the dual group of G) to admissible smooth representations of G(F). There is conjectured to be a finite-to-one map from irreducible smooth representations of G(F) to L-parameters, and there are many different approaches to parametrizing the fibers of such a map. 

 

The goal of this talk is to explain some of these approaches;  a special focus will be placed on the so-called "isocrystal" and "rigid" local Langlands correspondences. The former is best suited for building on the recent breakthroughs of Fargues-Scholze, while the latter is the broadest and is well-suited to endoscopy (a version of functoriality). We will discuss a proof of the equivalence of these two approaches, initiated by Kaletha for p-adic fields and extended to arbitrary nonarchimedean local fields in my recent work.