Categorical Langlands Summit

KIAS, Seoul

June 15-26, 2026

Organizers: Sug Woo Shin, David Hansen

Speakers:

Alexander Bertoloni Meli

Arnaud Eteve

Ian Gleason

David Hansen

David Helm

Dongryul Kim

Teruhisa Koshikawa

Lucas Mann

Pol van Hoften

Xiangqian Yang

Mingjia Zhang

Xinwen Zhu

Panelists: Linus Hamann, Naoki Imai, Jared Weinstein, Yifei Zhao

Goals. The local Langlands conjecture predicts a parametrization of smooth representations of a p-adic group in Galois-theoretic terms. In recent years, a conjectural but very powerful categorical upgrade of this story has emerged, which comes in two distinct variants as formulated by Zhu and Fargues-Scholze. These two versions of the conjecture have now been proved in many cases, in very recent papers of Zhu and Hansen-Mann.* The aim of this workshop is to explain these conjectures and give a detailed exposition of their (partial) proofs by Z. and H.-M., along with relevant background and applications, and to discuss future directions.

*The H.-M. paper will be available before the conference.

Overview.

Week one (June 15-19). Minicourses and lectures on relevant background and recent developments, including roughly the following:

1. Representation theory of reductive p-adic groups, including structure theory a la Bernstein, finiteness properties and dualities, the Bernstein center, etc.

2. Stacks of bundles, isocrystals, shtukas: their basic geometry; their sheaf theory.

3. An overview of Fargues-Scholze.

4. Review of the statement and status of “classical” local Langlands for GLn and classical groups after Harris-Taylor, Arthur, Moeglin, etc., including compatibility with Fargues-Scholze.

5. Categorical trace methods with applications & illustrations via Deligne-Lusztig theory.

6. Bezrukavnikov’s equivalence.

7. Moduli of L-parameters.

8. The work of Ben-Zvi-Chen-Helm-Nadler.

4 discussions x 1 hour

Week two (June 22-26). Most of this week will consist of two main lecture series (each 6 lectures):

Xinwen Zhu: Tame/unipotent categorical local Langlands

David Hansen and Lucas Mann: Categorical local Langlands for GL(n)

There will also be minicourses on the following topics:

1. Advanced theory of coherent sheaves on the L-parameter stack

2. The relation between the two forms of the conjecture

3. Arithmetic applications of the conjectures

Aside from the two main lecture series, the remaining talks will be distributed among the other confirmed speakers ahead of the workshop. We will also have multiple informal Q&A sessions and discussion sessions throughout the conference, which will be moderated by the panelists.

Expected schedule for both weeks:

M/Tu/Th/F: 2 morning lectures, 2 afternoon lectures, 1 Q&A or discussion session

W: 2 morning lectures, free afternoon

Registration: Click here.

Registration is required to attend this event. The deadline for registration is April 1, 2026.

Background.

In order to get to the latest developments, we are not going to start from zero in week one. We expect participants to come in with some familiarity with most of the following topics, which will not be reviewed in detail in the lectures (though some aspects of them may be quickly recalled):

• Algebraic number theory, commutative algebra, linear algebraic groups
• Algebraic geometry (scheme theory, Grothendieck topologies, derived categories of quasicoherent sheaves and ell-adic sheaves, stacky stuff)

• Adic spaces, perfectoid spaces, some exposure to diamonds and v-sheaves, the Fargues-Fontaine curve, cf. Berkeley Lecture Notes, AWS2017, Heidelberg spring school 2023 (IHES summer school lectures)
• Structure theory and smooth representation theory of p-adic groups (e.g., root data, parabolics, hyperspecial/Iwahori subgroups // definition of smooth, admissible, tempered, square-integrable, supercuspidal, parabolic induction, Jacquet module, Hecke algebra, Satake isomorphism, …) cf. Ottawa Lectures, AWS2025
• Basics of the local Langlands correspondence (e.g., L-groups, L-parameters, L-packets, the statement for GL(n), …) cf. Kaletha-Taibi, Kaletha, IHES summer school lectures
• Geometric Satake. The statement may be recalled in lectures, but in addition to the original papers, we refer the audience to existing lecture notes, cf. X. Zhu and Baumann—Riche, and videos for learning, cf. Williamson (colloquium), Scholze (mixed-char), Madapusi’s course.
• Category theory (including the (very) basics of infty-categories)

Further references.

Below is a list of references that participants are encouraged (though not assumed) to explore before the workshop.

• Bernstein center

• Bernstein (redige par Deligne), Le centre de Bernstein
• Bernstein (redige par Rumelhart), Representations of p-adic groups (unfinished 1992 lecture notes)

• Categorical traces and some more

• H. Chen, categorical traces
• Ganter—Kapranov, Representation and character theory in 2-categories
• X. Zhu, Geometric Satake, categorical traces, and arithmetic of Shimura varieties

• Moduli of Langlands parameters

• Dat—Helm—Kurinczuk—Moss, Moduli of Langlands parameters
• X. Zhu, Coherent sheaves on the stack of Langlands parameters (video)

• Geometrization or categorification of local Langlands correspondence (LLC)

• Ben-Zvi—Chen—Helm—Nadler, Between Coherent and Constructible Local Langlands Correspondences (video1, video2, video3)
• Fargues—Scholze, Geometrization of the local Langlands correspondence (Bonn lectures, IHES 2022, IHES 2017)
• Hansen, Beijing notes on the categorical Langlands conjecture (video)
• X. Zhu, Tame categorical local Langlands correspondence (video)
• ”McGill (Montreal) lectures”

• Comparison between Fargues—Scholze and X. Zhu for LLC

• Gleason—Ivanov—Zillinger, Meromorphic vector bundles on the Fargues—Fontaine curve (video)