Explicit Mod-Ell Categorical Local Langlands Correspondence for Depth-Zero Supercuspidal

Explore an advanced mathematics lecture on the explicit mod-ℓ categorical local Langlands correspondence for depth-zero supercuspidal representations of GL_2. Delve into the intricacies of non-archimedean local fields, moduli spaces of Langlands parameters, and categories of smooth representations. Examine the explicit description of connected components in the moduli space containing irreducible tame L-parameters and the corresponding blocks of smooth representations. Follow the speaker's focus on the GL_2 case for simplicity, while understanding the potential applicability to other split reductive groups. Gain insights into the abstract matching between the two sides and, time permitting, learn about the categorical local Langlands correspondence for depth-zero supercuspidal representations of GL_2 with Z_ℓ-bar coefficients in the Fargues-Scholze formulation.