Prof. David Hansen----Morningside Center of Mathematics Chinese Academy of Sciences

Classical and categorical local Langlands correspondences

Prof. David Hansen

2023-07-20 7.18, 7.20, 7.25, 7.27 10:00-11:30

MCM110&Online

Speaker: Prof. David Hansen (National University of Singapore)

Time: 10:00am-11:30am on July 18 & July 20 & July 25 & July 27

Place: MCM110 & ZOOM (Zoom ID: 332 983 6068 Password: mcm1234)

Title: Classical and categorical local Langlands correspondences

Abstract: The local Langlands conjecture (LLC) predicts that irreducible smooth representations of p-adic groups can be parametrized by Galois-theoretic data. A precise form of this prediction is now known for many groups, through the efforts of many people, via arguments of global nature. On the other hand, seminal recent work of Fargues--Scholze gives a completely general and purely local construction of a candidate for LLC, and in fact predicts a remarkable upgrade of the LLC to an equivalence of categories. The key philosophy in their work is that LLC should be reinterpreted as geometric Langlands over the stack of G-bundles on the Fargues--Fontaine curve.

In these lectures, I plan to formulate a number of new conjectures in the context of the Fargues--Scholze program. In particular, I will: give a precise unconditional formulation of the categorical local Langlands conjecture (with characteristic zero coefficients); formulate some expected properties of the categorical LLC and some unconditional conjectures which they naturally suggest; explain the precise relationship (known and conjectural) between the categorical and classical LLC.

I will quickly recall the key structures from Fargues--Scholze in the first lecture, but audience members are also strongly encouraged to (re)read the introduction of (https://arxiv.org/abs/2102.13459) beforehand.