categorical form of the local Langlands conjecture using the geometry on
a certain infinite dimensional stack classifying F-iscocrystals with
additional structure, analogous to a conjecture made by Fargues-Scholze.
Using the categorical trace construction we obtain the “unipotent part”
of the conjecture from Bezrukavnikov’s equivalence of two realizations
of the affine Hecke category. Joint work in progress with Xinwen Zhu.