# Summer School on the Langlands Program

Europe/Paris
Marilyn and James Simons Conference Center (IHES)

### Marilyn and James Simons Conference Center

#### IHES

35 route de Chartres, F-91440 Bures-sur-Yvette, France
Description

2022 IHES SUMMER SCHOOL

### Thank you for your understanding and cooperation!

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Organizing Committee: Pierre-Henri Chaudouard (IMJ-PRG),  Wee Teck Gan (National Univ. of Singapore), Tasho Kaletha (Univ. of Michigan), Yiannis Sakellaridis (Johns Hopkins Univ.)

Scientific Committee: Gérard Laumon (Univ. Paris-Sud), Colette Mœglin (IMJ-PRG), Bo Châu NGÔ (Chicago Univ.), Jean-Loup Waldspurger (IMJ-PRG)

The "Summer School on the Langlands Program" will be held at the Institut des Hautes Etudes Scientifiques (IHES) from 11 to 29 July. IHES is located in Bures-sur-Yvette, south of Paris (40 minutes by train from Paris).

This school is open to everybody but intended primarily for young participants, including Ph.D. students and postdoctoral fellows.

It has been almost 45 years since the influential summer school held in Corvallis, Oregon in 1977 brought together the leading experts of the Langlands program and defined the research agenda in this area for subsequent decades, at the same time inspiring and enabling several generations of young researchers to join in this exciting journey. This 3-week IHES summer school aims to do the same for the next phase of development in the Langlands program.

Recent decades have brought tremendous progress on the project of endoscopy, the extension of the Langlands program to the “relative” setting of spherical varieties and other related spaces, numerous successful “explicit” methods (such as the theta correspondence) to construct functoriality, novel ideas “beyond endoscopy”, and arithmetic applications of both the theta correspondence and the relative trace formula to the study of special cycles and their generating series. Ideas from the geometric Langlands program have begun impacting and enriching the classical Langlands program in significant ways. In particular, the idea that the “space of Langlands parameters” is not just a set, but a (putative) geometric space, can be used to organize a lot of developments around reciprocity, including the Taylor–Wiles method, derived structures, the Langlands correspondence over function fields, and the geometrization of the local Langlands conjecture.

The summer school will attempt to bring these exciting new directions together and explore their interactions.

SPEAKERS:

• BEN-ZVI David (UT Austin)
• BEUZART-PLESSIS Raphaël (Univ. Aix-Marseille)
• CARAIANI Ana (Imperial College)
• CHAUDOUARD Pierre-Henri (IMJ-PRG)
• DAT Jean-François (IMJ-PRG)
• EMERTON Matthew (Chicago Univ.)
• FARGUES Laurent (IMJ-PRG)
• FENG Tony (MIT)
• FINTZEN Jessica (Duke Univ. & Cambridge Univ.)
• GAN Wee Teck (National Univ. of Singapore)
• GEE Toby (Imperial College)
• HARRIS Michael (Columbia Univ.)
• HELLMANN Eugen (Univ. Münster)
• KALETHA Tasho (Univ. of Michigan)
• LAPID Erez (Weizmann Inst.)
• LI Chao (Columbia Univ.)
• MASON-BROWN Lucas (Univ. of Oxford)
• MOREL Sophie (ENS Lyon)
• NGÔ Bo Châu (Chicago Univ.)
• RASKIN Sam (Univ. of Texas)
• SAKELLARIDIS Yiannis (Johns Hopkins Univ.)
• SCHOLZE Peter (Univ. Bonn)
• SHIN Sug Woo  (UC Berkeley)
• TAÏBI Olivier (ENS Lyon)
• VENKATESH Akshay (IAS)
• WEINSTEIN Jared (Boston Univ.)
• XUE Cong (IMJ-PRG)
• YUN Zhiwei (MIT)
• ZHANG Wei (MIT)
• ZHU Xinwen (CALTECH)

This is an IHES Summer School, organized in partnership with the Clay Mathematical Institute, the National Science Foundation, the support of the Société Générale, the FMJH and Qube RT.

Contact: Elisabeth Jasserand
• Monday, July 11
• 9:00 AM 9:20 AM
Welcome coffee and registration 20m
• 9:20 AM 10:40 AM
The Local Langlands Conjecture (1/3) 1h 20m

We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.

Speaker: Prof. Olivier TAÏBI (ENS Lyon)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Some Perspectives on Eisenstein Series (1/2) 1h 20m

This is a review of some developments in the theory of the Eisenstein series since Corvallis.

Speaker: Prof. Erez LAPID (Weizmann Institute)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break (buffet at IHES) 2h
• 3:00 PM 4:00 PM
The Local Langlands Conjecture (2/3) 1h

We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.

Speaker: Prof. Olivier TAÏBI (ENS Lyon)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Shimura Varieties: outline (1/3) 1h

Depending on your point of view, Shimura varieties are a special kind of locally symmetric spaces, a generalization of moduli spaces of abelian schemes with extra structures, or the imperfect characteristic 0 version of moduli spaces of shtuka. They play an important role in the Langlands program because they have many symmetries (the Hecke correspondences) allowing us to link their cohomology to the theory of automorphic representations, and on the other hand, they are explicit enough for this cohomology to be computable. The goal of these lectures is to give an introduction to Shimura varieties, to present some examples, and to explain the conjectures on their cohomology (at least in the simplest case).

Speaker: Prof. Sophie MOREL (ENS Lyon)
• 5:20 PM 6:00 PM
Q&A 40m
• Tuesday, July 12
• 9:00 AM 9:20 AM
Welcome Coffee 20m
• 9:20 AM 11:00 AM
Introduction to the (Relative) Trace Formula (1/2) 1h 40m

The relative trace formula as envisioned by Jacquet and others is a possible generalization of the Arthur-Selberg trace formula. It is expected to be a useful tool in the relative Langlands program. We will try to present the general principle and give some examples and applications

Speaker: Prof. Pierre-Henri CHAUDOUARD (IMJ-PRG)
• 11:00 AM 11:20 AM
Coffee break 20m
• 11:20 AM 12:20 PM
Some Perspectives on Eisenstein Series (2/2) 1h

This is a review of some developments in the theory of the Eisenstein series since Corvallis.

Speaker: Prof. Erez LAPID (Weizmann Institute)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Shimura Varieties: outline (2/3) 1h

Depending on your point of view, Shimura varieties are a special kind of locally symmetric spaces, a generalization of moduli spaces of abelian schemes with extra structures, or the imperfect characteristic 0 version of moduli spaces of shtuka. They play an important role in the Langlands program because they have many symmetries (the Hecke correspondences) allowing us to link their cohomology to the theory of automorphic representations, and on the other hand, they are explicit enough for this cohomology to be computable. The goal of these lectures is to give an introduction to Shimura varieties, to present some examples, and to explain the conjectures on their cohomology (at least in the simplest case).

Speaker: Prof. Sophie MOREL (ENS Lyon)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
The Local Langlands Conjecture (3/3) 1h

We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.

Speaker: Prof. Olivier TAÏBI (ENS Lyon)
• 5:20 PM 6:00 PM
Q&A 40m
• Wednesday, July 13
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
Introduction to the (Relative) Trace Formula (2/2) 1h 20m

The relative trace formula as envisioned by Jacquet and others is a possible generalization of the Arthur-Selberg trace formula. It is expected to be a useful tool in the relative Langlands program. We will try to present the general principle and give some examples and applications.

Speaker: Prof. Pierre-Henri CHAUDOUARD (IMJ-PRG)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Arthur’s Conjectures and the Orbit Method for Real Reductive Groups 1h 20m

The most fundamental unsolved problem in the representation theory of Lie groups is the Problem of the Unitary Dual: given a reductive Lie group G, this problem asks for a parameterization of the set of irreducible unitary G-representations. There are two big "philosophies" for approaching this problem. The Orbit Method of Kostant and Kirillov seeks to parameterize irreducible unitary representations in terms of finite covers of co-adjoint G-orbits. Arthur's conjectures suggest a parameterization in terms of certain combinatorial gadgets (i.e. Arthur parameters) related to the Langlands dual group G^{\vee} of G.
In this talk, I will define these correspondences precisely in the case of complex groups. I will also define a natural duality map from Arthur parameters (for G^{\vee}) to co-adjoint covers (for G) which, in a certain precise sense, intertwines these correspondences. This talk is partially based on joint work with Ivan Losev and Dmitryo Matvieievskyi.

Speaker: Prof. Lucas MASON-BROWN (Univ. Oxford)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Introduction to Shtukas and their Moduli (1/3) 1h

We will start with basic definitions of Drinfeld Shtukas and their moduli stacks. Then we will talk about its geometric and cohomological properties and important constructions such as Hecke correspondences and partial Frobenius. We will also mention its relation with Drinfeld modules and analogy with motives.

Speaker: Prof. Zhiwei YUN (MIT)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Shimura Varieties: outline (3/3) 1h

Depending on your point of view, Shimura varieties are a special kind of locally symmetric spaces, a generalization of moduli spaces of abelian schemes with extra structures, or the imperfect characteristic 0 version of moduli spaces of shtuka. They play an important role in the Langlands program because they have many symmetries (the Hecke correspondences) allowing us to link their cohomology to the theory of automorphic representations, and on the other hand, they are explicit enough for this cohomology to be computable. The goal of these lectures is to give an introduction to Shimura varieties, to present some examples, and to explain the conjectures on their cohomology (at least in the simplest case).

Speaker: Prof. Sophie MOREL (ENS Lyon)
• 5:20 PM 6:00 PM
Q&A 40m
• Thursday, July 14
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
The Relative Langlands Program (1/3) 1h 20m

This is an introduction to what is nowadays called the Relative Langlands Program whose rough aim is to enhance the original Langlands conjectures from group to certain G-spaces named spherical varieties. The development of this relative aspect of the Langlands program originates from the discovery, by way of many examples, that automorphic periods and local distinction problems are often related to functoriality and/or L-functions.

Speaker: Prof. Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Coherent Sheaves on the Stack of Langlands Parameters (1/3) 1h 20m

I will give an impression of some recent new ideas appearing in the arithmetic Langlands program, with an emphasis on coherent sheaves on moduli spaces of Langlands parameters.

Speaker: Prof. Xinwen ZHU (Caltech)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Introduction to Shtukas and their Moduli (2/3) 1h

We will start with basic definitions of Drinfeld Shtukas and their moduli stacks. Then we will talk about its geometric and cohomological properties and important constructions such as Hecke correspondences and partial Frobenius. We will also mention its relation with Drinfeld modules and analogy with motives.

Speaker: Prof. Zhiwei YUN (MIT)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Introduction to Shtukas and their Moduli (3/3) 1h

We will start with basic definitions of Drinfeld Shtukas and their moduli stacks. Then we will talk about its geometric and cohomological properties and important constructions such as Hecke correspondences and partial Frobenius. We will also mention its relation with Drinfeld modules and analogy with motives.

Speaker: Prof. Zhiwei YUN (MIT)
• 5:20 PM 6:00 PM
Q&A 40m
• Friday, July 15
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
The Relative Langlands Program (2/3) 1h 20m

This is an introduction to what is nowadays called the Relative Langlands Program whose rough aim is to enhance the original Langlands conjectures from group to certain G-spaces named spherical varieties. The development of this relative aspect of the Langlands program originates from the discovery, by way of many examples, that automorphic periods and local distinction problems are often related to functoriality and/or L-functions.

Speaker: Prof. Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Coherent Sheaves on the Stack of Langlands Parameters (2/3) 1h 20m

I will give an impression of some recent new ideas appearing in the arithmetic Langlands program, with an emphasis on coherent sheaves on moduli spaces of Langlands parameters.

Speaker: Prof. Xinwen ZHU (Caltech)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Coherent Sheaves on the Stack of Langlands Parameters (3/3) 1h

I will give an impression of some recent new ideas appearing in the arithmetic Langlands program, with an emphasis on coherent sheaves on moduli spaces of Langlands parameters.

Speaker: Prof. Xinwen ZHU (Caltech)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
The Relative Langlands Program (3/3) 1h

This is an introduction to what is nowadays called the Relative Langlands Program whose rough aim is to enhance the original Langlands conjectures from group to certain G-spaces named spherical varieties. The development of this relative aspect of the Langlands program originates from the discovery, by way of many examples, that automorphic periods and local distinction problems are often related to functoriality and/or L-functions.

Speaker: Prof. Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
• 5:20 PM 6:00 PM
Q&A 40m
• Monday, July 18
• 9:00 AM 9:20 AM
Welcom coffee 20m
• 9:20 AM 10:40 AM
A Brief Introduction to the Trace Formula and its Stabilization (1/2) 1h 20m

We will discuss the derivation of the stable Arthur-Selberg trace formula. In the first lecture we will focus on anisotropic reductive groups, for which the trace formula can be derived easily. We will then discuss the stabilization of this trace formula, which is unconditional on the geometric side, and relies on the Arthur conjectures on the spectral side. In the second lecture we will sketch the case of an arbitrary reductive group, which causes many analytic difficulties. We will briefly describe the various stops on the road to the stable trace formula, including the coarse and fine expansions of the non-invariant trace formula, as well the invariant trace formula. Examples will be given for the group SL_2. Towards the end, we will discuss the application of the stable trace formula to the classification of representations of classical groups.

Speaker: Prof. Tasho KALETHA (Univ. Michigan)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Local Shtukas and the Langlands Program (1/2) 1h 20m

In the Langlands program over number fields, automorphic representations and Galois representations are placed into correspondence, using the cohomology of Shimura varieties as an intermediary. Over a function field, the appropriate intermediary is a moduli space of shtukas. We introduce the shtukas and their local analogs, which play a similar role in the local Langlands program. Along the way, we construct the Fargues-Fontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the lectures of Fargues-Scholze.

Speaker: Prof. Jared WEINSTEIN (Boston Univ.)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Orbital Integrals, Moduli Spaces and Invariant Theory (1/3) 1h

The goal of these lectures is to sketch a general framework to study orbital integrals over equal characteristic local fields by means of moduli spaces of Hitchin type following the main lines of the proof of the fundamental lemma for Lie algebras. After recalling basic elements of the proof of the fundamental lemma for Lie algebras as well as recent related developments, I will explain an invariant theoretic construction which should a basic tool to understand general orbital integrals.

Speaker: Prof. Bao Châu NGÔ (Chicago Univ.)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Orbital Integrals, Moduli Spaces and Invariant Theory (2/3) 1h

The goal of these lectures is to sketch a general framework to study orbital integrals over equal characteristic local fields by means of moduli spaces of Hitchin type following the main lines of the proof of the fundamental lemma for Lie algebras. After recalling basic elements of the proof of the fundamental lemma for Lie algebras as well as recent related developments, I will explain an invariant theoretic construction which should a basic tool to understand general orbital integrals.

Speaker: Prof. Bao Châu NGÔ (Chicago Univ.)
• 5:20 PM 6:00 PM
Q&A 40m
• Tuesday, July 19
• 9:00 AM 9:20 AM
Coffee 20m
• 9:20 AM 10:40 AM
Local Shtukas and the Langlands Program (2/2) 1h 20m

In the Langlands program over number fields, automorphic representations and Galois representations are placed into correspondence, using the cohomology of Shimura varieties as an intermediary. Over a function field, the appropriate intermediary is a moduli space of shtukas. We introduce the shtukas and their local analogs, which play a similar role in the local Langlands program. Along the way, we construct the Fargues-Fontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the lectures of Fargues-Scholze.

Speaker: Prof. Jared WEINSTEIN (Boston Univ.)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
A Brief Introduction to the Trace Formula and its Stabilization (2/2) 1h 20m

We will discuss the derivation of the stable Arthur-Selberg trace formula. In the first lecture we will focus on anisotropic reductive groups, for which the trace formula can be derived easily. We will then discuss the stabilization of this trace formula, which is unconditional on the geometric side, and relies on the Arthur conjectures on the spectral side. In the second lecture we will sketch the case of an arbitrary reductive group, which causes many analytic difficulties. We will briefly describe the various stops on the road to the stable trace formula, including the coarse and fine expansions of the non-invariant trace formula, as well the invariant trace formula. Examples will be given for the group SL_2. Towards the end, we will discuss the application of the stable trace formula to the classification of representations of classical groups.

Speaker: Prof. Tasho KALETHA (Univ. Michigan)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Cohomology Sheaves of Stacks of Shtukas (1/2) 1h

Cohomology sheaves and cohomology groups of stacks of shtukas are used in the Langlands program for function fields. We will explain (1) how the Eichler-Shimura relations imply the finiteness property of the cohomology groups, (2) how the finiteness and Drinfeld's lemma imply the action of the Weil group of the function field on the cohomology groups, and (3) how this action and the "Zorro lemma" imply the smoothness of the cohomology sheaves. The smoothness will be used in Sam Raskin’s lecture.

Speaker: Prof. Cong XUE (IMJ-PRG)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Cohomology Sheaves of Stacks of Shtukas (2/2) 1h

Cohomology sheaves and cohomology groups of stacks of shtukas are used in the Langlands program for function fields. We will explain (1) how the Eichler-Shimura relations imply the finiteness property of the cohomology groups, (2) how the finiteness and Drinfeld's lemma imply the action of the Weil group of the function field on the cohomology groups, and (3) how this action and the "Zorro lemma" imply the smoothness of the cohomology sheaves. The smoothness will be used in Sam Raskin’s lecture.

Speaker: Prof. Cong XUE (IMJ-PRG)
• 5:20 PM 6:00 PM
Q&A 40m
• Wednesday, July 20
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
An Introduction to the Categorical p-adic Langlands Program (1/4) 1h 20m

An introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach'' and "analytic'' settings.

Speakers: Prof. Eugen HELLMANN (Univ. Münster) , Prof. Matthew EMERTON (Chicago Univ.) , Prof. Toby GEE (Imperial College)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Orbital Integrals, Moduli Spaces and Invariant Theory (3/3) 1h 20m

The goal of these lectures is to sketch a general framework to study orbital integrals over equal characteristic local fields by means of moduli spaces of Hitchin type following the main lines of the proof of the fundamental lemma for Lie algebras. After recalling basic elements of the proof of the fundamental lemma for Lie algebras as well as recent related developments, I will explain an invariant theoretic construction which should a basic tool to understand general orbital integrals.

Speaker: Prof. Bao Châu NGÔ
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
What Does Geometric Langlands Mean to a Number Theorist? (1/2) 1h
Speaker: Prof. Sam RASKIN (Univ. Texas)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
What Does Geometric Langlands Mean to a Number Theorist? (2/2) 1h
Speaker: Prof. Sam RASKIN (Univ. Texas)
• 5:20 PM 6:00 PM
Q&A 40m
• 6:00 PM 7:00 PM
Panel Discussion with Participants of the Corvallis Conference 1h

J. Arthur (Univ. Toronto), B. Casselman (Univ. of British Columbia), B. Gross (Harvard Univ.), M. Harris (Columbia Univ.), G. Henniart (Univ. Paris-Saclay), H. Jacquet (Columbia Univ.), J.P. Labesse (Aix-Marseille Université), K. Ribet (Berkeley Univ.), C. Soulé (CNRS-IHES), M.F. Vignéras (IMJ-PRG).

• Thursday, July 21
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
The Langlands Program and the Moduli of Bundles on the Curve (1/3) 1h 20m

I will speak about my joint work on the geometrization of the local Langlands correspondence.

Speakers: Prof. Laurent FARGUES (IMJ-PRG) , Prof. Peter SCHOLZE (Univ. Bonn)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Explicit Constructions of Automorphic Forms (1/2) 1h 20m

I will discuss the theory of theta correspondence, highlighting basic principles and recent results, before explaining how theta correspondence can now be viewed as part of the relative Langlands program.
I will then discuss other methods of construction of automorphic forms, such as automorphic descent and its variants and the generalized doubling method.

Speaker: Prof. Wee Teck GAN (Nat. Univ. Singapour)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
An Introduction to the Categorical p-adic Langlands Program (2/4) 1h

An introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach'' and "analytic'' settings.

Speakers: Prof. Eugen HELLMANN (Univ. Münster) , Prof. Matthew EMERTON (Chicago Univ.) , Prof. Toby GEE (Imperial College)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
An Introduction to the Categorical p-adic Langlands Program (3/4) 1h

An introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach'' and "analytic'' settings.

Speakers: Prof. Eugen HELLMANN (Univ. Münster) , Prof. Matthew EMERTON (Chicago Univ.) , Prof. Toby GEE (Imperial College)
• 5:20 PM 6:00 PM
Q&A 40m
• Friday, July 22
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
On Moduli Spaces of Local Langlands Parameters (1/2) 1h 20m

The moduli space of local Langlands parameters plays a key role in the formulation of some recent enhancements of the original local Langlands correspondence, such as the "local Langlands correspondence in families" and various "categorifications/geometrizations of LLC". We will explain their construction and basic properties, with special emphasis on the coarse moduli spaces.

Speaker: Prof. Jean-François DAT (IMJ-PRG)
• 10:40 AM 11:00 AM
Coffe break 20m
• 11:00 AM 12:20 PM
Explicit Constructions of Automorphic Forms (2/2) 1h 20m

I will discuss the theory of theta correspondence, highlighting basic principles and recent results, before explaining how theta correspondence can now be viewed as part of the relative Langlands program.
I will then discuss other methods of construction of automorphic forms, such as automorphic descent and its variants and the generalized doubling method.

Speaker: Prof. Wee Teck GAN (Nat. Univ. Singapour)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Supercuspidal Representations: Construction, Classification, and Characters (1/2) 1h

We have seen in the first week of the summer school that the buildings blocks for irreducible representations of p-adic groups are the supercuspidal representations. In these talks we will explore explicit exhaustive constructions of these supercuspidal representations and their character formulas and observe a striking parallel between a large class of these representations in the p-adic world and discrete series representations of real algebraic Lie groups. A key ingredient for the construction of supercuspidal representations is the Bruhat--Tits theory and Moy--Prasad filtration, which we will introduce in the lecture series.

Speaker: Prof. Jessica FINTZEN (Duke Univ. & Cambridge Univ.)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
The Langlands Program and the Moduli of Bundles on the Curve (2/3) 1h

I will speak about my joint work on the geometrization of the local Langlands correspondence.

Speakers: Prof. Laurent FARGUES (IMJ-PRG) , Prof. Peter SCHOLZE (Univ. Bonn)
• 5:20 PM 6:00 PM
Q&A 40m
• Monday, July 25
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
An Introduction to the Categorical p-adic Langlands Program (4/4) 1h 20m

An introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach'' and "analytic'' settings.

Speakers: Prof. Eugen HELLMANN (Univ. Münster) , Prof. Matthew EMERTON (Chicago Univ.) , Prof. Toby GEE (Imperial College)
• 10:40 AM 11:00 AM
Coffe break 20m
• 11:00 AM 12:20 PM
Supercuspidal Representations: Construction, Classification, and Characters (2/2) 1h 20m

We have seen in the first week of the summer school that the buildings blocks for irreducible representations of p-adic groups are the supercuspidal representations. In these talks we will explore explicit exhaustive constructions of these supercuspidal representations and their character formulas and observe a striking parallel between a large class of these representations in the p-adic world and discrete series representations of real algebraic Lie groups. A key ingredient for the construction of supercuspidal representations is the Bruhat--Tits theory and Moy--Prasad filtration, which we will introduce in the lecture series.

Speaker: Prof. Jessica FINTZEN (Duke Univ. & Cambridge Univ.)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Branching Laws: Homological Aspects 1h

By this time in the Summer School, the audience will have seen the question about decomposing a group's representation when restricted to a subgroup, referred to as the branching law. In this lecture, we focus attention on homological aspects of the branching law. The lecture will survey this topic beginning from the beginning and going up to several results which have recently been proved.

Speaker: Prof. Dipendra PRASAD (IIT Bombay)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
On Moduli Spaces of Local Langlands Parameters (2/2) 1h

The moduli space of local Langlands parameters plays a key role in the formulation of some recent enhancements of the original local Langlands correspondence, such as the "local Langlands correspondence in families" and various "categorifications/geometrizations of LLC". We will explain their construction and basic properties, with special emphasis on the coarse moduli spaces.

Speaker: Prof. Jean-François DAT (IMJ-PRG)
• 5:20 PM 6:00 PM
Q&A 40m
• Tuesday, July 26
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
The Langlands Program and the Moduli of Bundles on the Curve (3/3) 1h 20m

I will speak about my joint work on the geometrization of the local Langlands correspondence.

Speakers: Prof. Laurent FARGUES (IMJ-PRG) , Prof. Peter SCHOLZE (Univ. Bonn)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:00 PM
Shimura Varieties and Modularity (1/3) 1h

We describe the construction of Galois representations associated to regular algebraic cuspidal automorphic representations of GL_n over a CM field, as well as those Galois representations associated to torsion classes that occur in the Betti cohomology of the corresponding locally symmetric spaces. The emphasis will be on Scholze’s proof, which applies to torsion classes and which uses perfectoid Shimura varieties and the Hodge-Tate period morphism.

Speakers: Prof. Ana CARAIANI (Imperial College) , Prof. Sug Woo SHIN (UC Berkeley)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Shimura Varieties and Modularity (2/3) 1h

We describe the Calegari-Geraghty method for proving modularity lifting theorems beyond the classical setting of the Taylor-Wiles method. We discuss the three conjectures that this method relies on (existence of Galois representations, local-global compatibility and vanishing of cohomology outside a certain range of degrees) and their current status, and then explain the commutative algebra underlying the method.

Speakers: Prof. Ana CARAIANI (Imperial College) , Prof. Sug Woo SHIN (UC Berkeley)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Geometric and Arithmetic Theta Correspondences (1/2) 1h

Geometric/arithmetic theta correspondences provide correspondences between automorphic forms and cohomology classes/algebraic cycles on Shimura varieties. I will give an introduction focusing on the example of unitary groups and highlight recent advances in the arithmetic theory (also known as the Kudla program) and their applications.

Speaker: Prof. Chao LI (Columbia Univ.)
• 5:20 PM 6:00 PM
Q&A 40m
• Wednesday, July 27
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
Shimura Varieties and Modularity (3/3) 1h 20m

We discuss vanishing theorems for the cohomology of Shimura varieties with torsion coefficients, under a genericity condition at an auxiliary prime. We describe two complementary approaches to these results, due to Caraiani-Scholze and Koshikawa, both of which rely on the geometry of the Hodge-Tate period morphism for the corresponding Shimura varieties. Finally, we explain how these vanishing results can be applied to local-global compatibility questions for the Galois representations constructed in the first lecture.

Speakers: Prof. Ana CARAIANI (Imperial College) , Prof. Sug Woo SHIN (UC Berkeley)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Geometric and Arithmetic Theta Correspondences (2/2) 1h 20m

Geometric/arithmetic theta correspondences provide correspondences between automorphic forms and cohomology classes/algebraic cycles on Shimura varieties. I will give an introduction focusing on the example of unitary groups and highlight recent advances in the arithmetic theory (also known as the Kudla program) and their applications.

Speaker: Prof. Chao LI (Columbia Univ.)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Hamiltonian Actions and Langlands Duality (1/2) 1h

I will give a gentle introduction to my joint work with Ben-Zvi and Sakellaridis, in which we seek to formulate various phenomena in the Langlands program in terms of Hamiltonian actions of reductive groups. In particular, this makes visible a duality underlying the relative Langlands program.

Speaker: Prof. Akshay VENKATESH (IAS)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Hamiltonian Actions and Langlands Duality (2/2) 1h

I will give a gentle introduction to my joint work with Ben-Zvi and Sakellaridis, in which we seek to formulate various phenomena in the Langlands program in terms of Hamiltonian actions of reductive groups. In particular, this makes visible a duality underlying the relative Langlands program.

Speaker: Prof. Akshay VENKATESH (IAS)
• 5:20 PM 6:00 PM
Q&A 40m
• Thursday, July 28
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
High-dimensional Gross–Zagier Formula (1/2) 1h 20m

I'll discuss various generalizations of the Gross--Zagier formula to high-dimensional Shimura varieties, with an emphasis on the AGGP conjecture and the relative trace formula approach. Roughly the first lecture will be devoted to the global aspect and the second one to the local aspect.

Speaker: Prof. Wei ZHANG (MIT)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
Between Coherent and Constructible Local Langlands Correspondences 1h 20m

(Joint with Harrison Chen, David Helm, and David Nadler.)
Refined forms of the local Langlands correspondence seek to relate representations of reductive groups over local fields with sheaves on stacks of Langlands parameters. But what kind of sheaves? Conjectures in the spirit of Kazhdan-Lusztig theory describe representations of a group and its pure inner forms with fixed central character in terms of constructible sheaves. Conjectures in the spirit of geometric Langlands describe representations with varying central character of a large family of groups associated to isocrystals in terms of coherent sheaves. The latter conjectures also take place on a larger parameter space, in which Frobenius (or complex conjugation) is allowed a unipotent part.
In this talk, we propose a general mechanism that interpolates between these two settings. This mechanism derives from the theory of cyclic homology, as interpreted through circle actions in derived algebraic geometry. We apply this perspective to categorical forms of the local Langlands conjectures for both archimedean and non-archimedean local fields. In the archimedean case, we explain a conjectural realization of coherent local Langlands as geometric Langlands on the twistor line, the real counterpart of the Fargues-Fontaine curve, and its relation to constructible local Langlands via circle actions. In the non-archimedean case, we describe how circle actions relate coherent and constructible realizations of affine Hecke algebras and of all smooth representations of $GL_n$, and propose a mechanism to relate the two settings in general.

Speaker: Prof. David BEN-ZVI (UT Austin)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Buffet Lunch at IHES 2h
• 3:00 PM 4:00 PM
Derived Aspects of the Langlands Program (1/3) 1h

We discuss ways in which derived structures have recently emerged in connection with the Langlands correspondence, with an emphasis on derived Galois deformation rings and derived Hecke algebras.

Speakers: Prof. Michael HARRIS (Columbia Univ.) , Prof. Tony FENG (MIT)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Local and Global Questions “Beyond Endoscopy” (1/2) 1h

The near completion of the program of endoscopy poses the question of what lies next.
These talks will take a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship between local and global aspects. Central among those ideas is the one proposed in a 2000 lecture of R.~P.~Langlands, aiming to extract from the stable trace formula of a group $G$ the bulk of those automorphic representations in the image of the conjectural functorial lift corresponding to a morphism of $L$-groups ${^LH}\to {^LG}$. With the extension of the problem of functionality to the "relative'' setting of spherical varieties and related spaces, some structure behind such comparisons has started to reveal itself. In a seemingly unrelated direction, a program initiated by Braverman--Kazhdan, also around 2000, to generalize the Godement--Jacquet proof of the functional equation to arbitrary $L$-functions, has received renewed attention in recent years. We survey ideas and developments in this direction, as well, and discuss the relationship between the two programs.

Speaker: Prof. Yiannis SAKELLARIDIS (Johns Hopkins Univ.)
• 5:20 PM 6:00 PM
Q&A 40m
• Friday, July 29
• 9:00 AM 9:20 AM
Welcome coffee 20m
• 9:20 AM 10:40 AM
Derived Aspects of the Langlands Program (2/3) 1h 20m

We discuss ways in which derived structures have recently emerged in connection with the Langlands correspondence, with an emphasis on derived Galois deformation rings and derived Hecke algebras.

Speakers: Prof. Michael HARRIS (Columbia Univ.) , Prof. Tony FENG (MIT)
• 10:40 AM 11:00 AM
Coffee break 20m
• 11:00 AM 12:20 PM
High-dimensional Gross–Zagier Formula (2/2) 1h 20m

I'll discuss various generalizations of the Gross--Zagier formula to high-dimensional Shimura varieties, with an emphasis on the AGGP conjecture and the relative trace formula approach. Roughly the first lecture will be devoted to the global aspect and the second one to the local aspect.

Speaker: Prof. Wei ZHANG (MIT)
• 12:30 PM 1:00 PM
Q&A 30m
• 1:00 PM 3:00 PM
Lunch break 2h
• 3:00 PM 4:00 PM
Local and Global Questions “Beyond Endoscopy” (2/2) 1h

The near completion of the program of endoscopy poses the question of what lies next.
These talks will take a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship between local and global aspects. Central among those ideas is the one proposed in a 2000 lecture of R.~P.~Langlands, aiming to extract from the stable trace formula of a group $G$ the bulk of those automorphic representations in the image of the conjectural functorial lift corresponding to a morphism of $L$-groups ${^LH}\to {^LG}$. With the extension of the problem of functionality to the "relative'' setting of spherical varieties and related spaces, some structure behind such comparisons has started to reveal itself. In a seemingly unrelated direction, a program initiated by Braverman--Kazhdan, also around 2000, to generalize the Godement--Jacquet proof of the functional equation to arbitrary $L$-functions, has received renewed attention in recent years. We survey ideas and developments in this direction, as well, and discuss the relationship between the two programs.

Speaker: Prof. Yiannis SAKELLARIDIS (Johns Hopkins Univ.)
• 4:00 PM 4:20 PM
Coffee break 20m
• 4:20 PM 5:20 PM
Derived Aspects of the Langlands Program (3/3) 1h

We discuss ways in which derived structures have recently emerged in connection with the Langlands correspondence, with an emphasis on derived Galois deformation rings and derived Hecke algebras.

Speakers: Prof. Michael HARRIS (Columbia Univ.) , Prof. Tony FENG (MIT)
• 5:20 PM 6:00 PM
Q&A 40m