Name: Summer School on the Langlands Program
Start: 2022-07-11T08:30:00+02:00
End: 2022-07-29T19:00:00+02:00
Location: IHES

Monday, July 11, 2022

9:00 AM Welcome coffee and registration
Welcome coffee and registration
9:00 AM - 9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM The Local Langlands Conjecture (1/3) - Olivier TAÏBI (ENS Lyon)
The Local Langlands Conjecture (1/3)
- Olivier TAÏBI (ENS Lyon)
9:20 AM - 10:40 AM
Room: Marilyn and James Simons Conference Center We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.
10:40 AM Coffee break
Coffee break
10:40 AM - 11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM Some Perspectives on Eisenstein Series (1/2) - Erez LAPID (Weizmann Institute)
Some Perspectives on Eisenstein Series (1/2)
- Erez LAPID (Weizmann Institute)
11:00 AM - 12:20 PM
Room: Marilyn and James Simons Conference Center This is a review of some developments in the theory of the Eisenstein series since Corvallis.
12:30 PM Q&A
Q&A
12:30 PM - 1:00 PM
Room: Marilyn and James Simons Conference Center

1:00 PM Lunch break (buffet at IHES)
Lunch break (buffet at IHES)
1:00 PM - 3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM The Local Langlands Conjecture (2/3) - Olivier TAÏBI (ENS Lyon)
The Local Langlands Conjecture (2/3)
- Olivier TAÏBI (ENS Lyon)
3:00 PM - 4:00 PM
Room: Marilyn and James Simons Conference Center We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.
4:00 PM Coffee break
Coffee break
4:00 PM - 4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM Shimura Varieties: outline (1/3) - Sophie MOREL (ENS Lyon)
Shimura Varieties: outline (1/3)
- Sophie MOREL (ENS Lyon)
4:20 PM - 5:20 PM
Room: Marilyn and James Simons Conference Center 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).
5:20 PM Q&A
Q&A
5:20 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
Tuesday, July 12, 2022

9:00 AM Welcome Coffee
Welcome Coffee
9:00 AM - 9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM Introduction to the (Relative) Trace Formula (1/2) - Pierre-Henri CHAUDOUARD (IMJ-PRG)
Introduction to the (Relative) Trace Formula (1/2)
- Pierre-Henri CHAUDOUARD (IMJ-PRG)
9:20 AM - 11:00 AM
Room: Marilyn and James Simons Conference Center 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
11:00 AM Coffee break
Coffee break
11:00 AM - 11:20 AM
Room: Marilyn and James Simons Conference Center
11:20 AM Some Perspectives on Eisenstein Series (2/2) - Erez LAPID (Weizmann Institute)
Some Perspectives on Eisenstein Series (2/2)
- Erez LAPID (Weizmann Institute)
11:20 AM - 12:20 PM
Room: Marilyn and James Simons Conference Center This is a review of some developments in the theory of the Eisenstein series since Corvallis.
12:30 PM Q&A
Q&A
12:30 PM - 1:00 PM
Room: Marilyn and James Simons Conference Center

1:00 PM Lunch break
Lunch break
1:00 PM - 3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM Shimura Varieties: outline (2/3) - Sophie MOREL (ENS Lyon)
Shimura Varieties: outline (2/3)
- Sophie MOREL (ENS Lyon)
3:00 PM - 4:00 PM
Room: Marilyn and James Simons Conference Center 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).
4:00 PM Coffee break
Coffee break
4:00 PM - 4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM The Local Langlands Conjecture (3/3) - Olivier TAÏBI (ENS Lyon)
The Local Langlands Conjecture (3/3)
- Olivier TAÏBI (ENS Lyon)
4:20 PM - 5:20 PM
Room: Marilyn and James Simons Conference Center We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.
5:20 PM Q&A
Q&A
5:20 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
Wednesday, July 13, 2022

9:00 AM Welcome coffee
Welcome coffee
9:00 AM - 9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM Introduction to the (Relative) Trace Formula (2/2) - Pierre-Henri CHAUDOUARD (IMJ-PRG)
Introduction to the (Relative) Trace Formula (2/2)
- Pierre-Henri CHAUDOUARD (IMJ-PRG)
9:20 AM - 10:40 AM
Room: Marilyn and James Simons Conference Center 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.
10:40 AM Coffee break
Coffee break
10:40 AM - 11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM Arthur’s Conjectures and the Orbit Method for Real Reductive Groups - Lucas MASON-BROWN (Univ. Oxford)
Arthur’s Conjectures and the Orbit Method for Real Reductive Groups
- Lucas MASON-BROWN (Univ. Oxford)
11:00 AM - 12:20 PM
Room: Marilyn and James Simons Conference Center 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.
12:30 PM Q&A
Q&A
12:30 PM - 1:00 PM
Room: Marilyn and James Simons Conference Center

1:00 PM Lunch break
Lunch break
1:00 PM - 3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM Introduction to Shtukas and their Moduli (1/3) - Zhiwei YUN (MIT)
Introduction to Shtukas and their Moduli (1/3)
- Zhiwei YUN (MIT)
3:00 PM - 4:00 PM
Room: Marilyn and James Simons Conference Center 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.
4:00 PM Coffee break
Coffee break
4:00 PM - 4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM Shimura Varieties: outline (3/3) - Sophie MOREL (ENS Lyon)
Shimura Varieties: outline (3/3)
- Sophie MOREL (ENS Lyon)
4:20 PM - 5:20 PM
Room: Marilyn and James Simons Conference Center 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).
5:20 PM Q&A
Q&A
5:20 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
Thursday, July 14, 2022

9:00 AM Welcome coffee
Welcome coffee
9:00 AM - 9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM The Relative Langlands Program (1/3) - Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
The Relative Langlands Program (1/3)
- Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
9:20 AM - 10:40 AM
Room: Marilyn and James Simons Conference Center 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.
10:40 AM Coffee break
Coffee break
10:40 AM - 11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM Coherent Sheaves on the Stack of Langlands Parameters (1/3) - Xinwen ZHU (Caltech)
Coherent Sheaves on the Stack of Langlands Parameters (1/3)
- Xinwen ZHU (Caltech)
11:00 AM - 12:20 PM
Room: Marilyn and James Simons Conference Center 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.
12:30 PM Q&A
Q&A
12:30 PM - 1:00 PM
Room: Marilyn and James Simons Conference Center

1:00 PM Lunch break
Lunch break
1:00 PM - 3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM Introduction to Shtukas and their Moduli (2/3) - Zhiwei YUN (MIT)
Introduction to Shtukas and their Moduli (2/3)
- Zhiwei YUN (MIT)
3:00 PM - 4:00 PM
Room: Marilyn and James Simons Conference Center 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.
4:00 PM Coffee break
Coffee break
4:00 PM - 4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM Introduction to Shtukas and their Moduli (3/3) - Zhiwei YUN (MIT)
Introduction to Shtukas and their Moduli (3/3)
- Zhiwei YUN (MIT)
4:20 PM - 5:20 PM
Room: Marilyn and James Simons Conference Center 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.
5:20 PM Q&A
Q&A
5:20 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
Friday, July 15, 2022

9:00 AM Welcome coffee
Welcome coffee
9:00 AM - 9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM The Relative Langlands Program (2/3) - Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
The Relative Langlands Program (2/3)
- Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
9:20 AM - 10:40 AM
Room: Marilyn and James Simons Conference Center 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.
10:40 AM Coffee break
Coffee break
10:40 AM - 11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM Coherent Sheaves on the Stack of Langlands Parameters (2/3) - Xinwen ZHU (Caltech)
Coherent Sheaves on the Stack of Langlands Parameters (2/3)
- Xinwen ZHU (Caltech)
11:00 AM - 12:20 PM
Room: Marilyn and James Simons Conference Center 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.
12:30 PM Q&A
Q&A
12:30 PM - 1:00 PM
Room: Marilyn and James Simons Conference Center

1:00 PM Lunch break
Lunch break
1:00 PM - 3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM Coherent Sheaves on the Stack of Langlands Parameters (3/3) - Xinwen ZHU (Caltech)
Coherent Sheaves on the Stack of Langlands Parameters (3/3)
- Xinwen ZHU (Caltech)
3:00 PM - 4:00 PM
Room: Marilyn and James Simons Conference Center 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.
4:00 PM Coffee break
Coffee break
4:00 PM - 4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM The Relative Langlands Program (3/3) - Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
The Relative Langlands Program (3/3)
- Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
4:20 PM - 5:20 PM
Room: Marilyn and James Simons Conference Center 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.
5:20 PM Q&A
Q&A
5:20 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
Saturday, July 16, 2022
Sunday, July 17, 2022