Talk 5: On the local Langlands parametrization for $G(F_q((t)))$

I will report on several recent results on the Langlands parametrization of irreducible representations of groups over local fields of positive characteristic.  The parametrization is defined algebraically in the work of V. Lafforgue, Genestier-Lafforgue, and Fargues-Scholze, using variants of the theory of moduli of vector bundles on curves; however, the results I will discuss are primarily obtained by analytic methods.  I will specifically report on results on ramification of local parameters (joint with Gan and Sawin), on the generalized Ramanujan conjecture (joint with Ciubotaru), and on joint work in progress with Beuzart-Plessis and Thorne on a strategy to invert the parametrization.