Summer School on the Langlands Program
from
Monday, July 11, 2022 (8:30 AM)
to
Friday, July 29, 2022 (7:00 PM)
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 Lpackets.
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 Lpackets.
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)

PierreHenri CHAUDOUARD
(
IMJPRG
)
Introduction to the (Relative) Trace Formula (1/2)
PierreHenri CHAUDOUARD
(
IMJPRG
)
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 ArthurSelberg 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 Lpackets.
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)

PierreHenri CHAUDOUARD
(
IMJPRG
)
Introduction to the (Relative) Trace Formula (2/2)
PierreHenri CHAUDOUARD
(
IMJPRG
)
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 ArthurSelberg 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 MASONBROWN
(
Univ. Oxford
)
Arthur’s Conjectures and the Orbit Method for Real Reductive Groups
Lucas MASONBROWN
(
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 Grepresentations. 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 coadjoint Gorbits. 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 coadjoint 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 BEUZARTPLESSIS
(
Univ. AixMarseille
)
The Relative Langlands Program (1/3)
Raphaël BEUZARTPLESSIS
(
Univ. AixMarseille
)
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 Gspaces 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 Lfunctions.
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 BEUZARTPLESSIS
(
Univ. AixMarseille
)
The Relative Langlands Program (2/3)
Raphaël BEUZARTPLESSIS
(
Univ. AixMarseille
)
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 Gspaces 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 Lfunctions.
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 BEUZARTPLESSIS
(
Univ. AixMarseille
)
The Relative Langlands Program (3/3)
Raphaël BEUZARTPLESSIS
(
Univ. AixMarseille
)
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 Gspaces 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 Lfunctions.
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
Monday, July 18, 2022
9:00 AM
Welcom coffee
Welcom coffee
9:00 AM  9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM
A Brief Introduction to the Trace Formula and its Stabilization (1/2)

Tasho KALETHA
(
Univ. Michigan
)
A Brief Introduction to the Trace Formula and its Stabilization (1/2)
Tasho KALETHA
(
Univ. Michigan
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
We will discuss the derivation of the stable ArthurSelberg 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 noninvariant 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.
10:40 AM
Coffee break
Coffee break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Local Shtukas and the Langlands Program (1/2)

Jared WEINSTEIN
(
Boston Univ.
)
Local Shtukas and the Langlands Program (1/2)
Jared WEINSTEIN
(
Boston Univ.
)
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
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 FarguesFontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the lectures of FarguesScholze.
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
Orbital Integrals, Moduli Spaces and Invariant Theory (1/3)

Bao Châu NGÔ
(
Chicago Univ.
)
Orbital Integrals, Moduli Spaces and Invariant Theory (1/3)
Bao Châu NGÔ
(
Chicago Univ.
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
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.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
Orbital Integrals, Moduli Spaces and Invariant Theory (2/3)

Bao Châu NGÔ
(
Chicago Univ.
)
Orbital Integrals, Moduli Spaces and Invariant Theory (2/3)
Bao Châu NGÔ
(
Chicago Univ.
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
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.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
Tuesday, July 19, 2022
9:00 AM
Coffee
Coffee
9:00 AM  9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM
Local Shtukas and the Langlands Program (2/2)

Jared WEINSTEIN
(
Boston Univ.
)
Local Shtukas and the Langlands Program (2/2)
Jared WEINSTEIN
(
Boston Univ.
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
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 FarguesFontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the lectures of FarguesScholze.
10:40 AM
Coffee break
Coffee break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
A Brief Introduction to the Trace Formula and its Stabilization (2/2)

Tasho KALETHA
(
Univ. Michigan
)
A Brief Introduction to the Trace Formula and its Stabilization (2/2)
Tasho KALETHA
(
Univ. Michigan
)
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
We will discuss the derivation of the stable ArthurSelberg 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 noninvariant 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.
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
Cohomology Sheaves of Stacks of Shtukas (1/2)

Cong XUE
(
IMJPRG
)
Cohomology Sheaves of Stacks of Shtukas (1/2)
Cong XUE
(
IMJPRG
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
Cohomology sheaves and cohomology groups of stacks of shtukas are used in the Langlands program for function fields. We will explain (1) how the EichlerShimura 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.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
Cohomology Sheaves of Stacks of Shtukas (2/2)

Cong XUE
(
IMJPRG
)
Cohomology Sheaves of Stacks of Shtukas (2/2)
Cong XUE
(
IMJPRG
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
Cohomology sheaves and cohomology groups of stacks of shtukas are used in the Langlands program for function fields. We will explain (1) how the EichlerShimura 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.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
Wednesday, July 20, 2022
9:00 AM
Welcome coffee
Welcome coffee
9:00 AM  9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM
An Introduction to the Categorical padic Langlands Program (1/4)

Toby GEE
(
Imperial College
)
Matthew EMERTON
(
Chicago Univ.
)
Eugen HELLMANN
(
Univ. Münster
)
An Introduction to the Categorical padic Langlands Program (1/4)
Toby GEE
(
Imperial College
)
Matthew EMERTON
(
Chicago Univ.
)
Eugen HELLMANN
(
Univ. Münster
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
An introduction to the "categorical" approach to the padic Langlands program, in both the "Banach'' and "analytic'' settings.
10:40 AM
Coffee break
Coffee break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Orbital Integrals, Moduli Spaces and Invariant Theory (3/3)

Bao Châu NGÔ
Orbital Integrals, Moduli Spaces and Invariant Theory (3/3)
Bao Châu NGÔ
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
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.
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
What Does Geometric Langlands Mean to a Number Theorist? (1/2)

Sam RASKIN
(
Univ. Texas
)
What Does Geometric Langlands Mean to a Number Theorist? (1/2)
Sam RASKIN
(
Univ. Texas
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
What Does Geometric Langlands Mean to a Number Theorist? (2/2)

Sam RASKIN
(
Univ. Texas
)
What Does Geometric Langlands Mean to a Number Theorist? (2/2)
Sam RASKIN
(
Univ. Texas
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
6:00 PM
Panel Discussion with Participants of the Corvallis Conference
Panel Discussion with Participants of the Corvallis Conference
6:00 PM  7:00 PM
Room: Marilyn and James Simons Conference Center
J. Arthur (Univ. Toronto), B. Casselman (Univ. of British Columbia), B. Gross (Harvard Univ.), M. Harris (Columbia Univ.), G. Henniart (Univ. ParisSaclay), H. Jacquet (Columbia Univ.), J.P. Labesse (AixMarseille Université), K. Ribet (Berkeley Univ.), C. Soulé (CNRSIHES), M.F. Vignéras (IMJPRG).
Thursday, July 21, 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 Langlands Program and the Moduli of Bundles on the Curve (1/3)

Laurent FARGUES
(
IMJPRG
)
Peter SCHOLZE
(
Univ. Bonn
)
The Langlands Program and the Moduli of Bundles on the Curve (1/3)
Laurent FARGUES
(
IMJPRG
)
Peter SCHOLZE
(
Univ. Bonn
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
I will speak about my joint work on the geometrization of the local Langlands correspondence.
10:40 AM
Coffee break
Coffee break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Explicit Constructions of Automorphic Forms (1/2)

Wee Teck GAN
(
Nat. Univ. Singapour
)
Explicit Constructions of Automorphic Forms (1/2)
Wee Teck GAN
(
Nat. Univ. Singapour
)
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
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.
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
An Introduction to the Categorical padic Langlands Program (2/4)

Toby GEE
(
Imperial College
)
Matthew EMERTON
(
Chicago Univ.
)
Eugen HELLMANN
(
Univ. Münster
)
An Introduction to the Categorical padic Langlands Program (2/4)
Toby GEE
(
Imperial College
)
Matthew EMERTON
(
Chicago Univ.
)
Eugen HELLMANN
(
Univ. Münster
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
An introduction to the "categorical" approach to the padic Langlands program, in both the "Banach'' and "analytic'' settings.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
An Introduction to the Categorical padic Langlands Program (3/4)

Toby GEE
(
Imperial College
)
Matthew EMERTON
(
Chicago Univ.
)
Eugen HELLMANN
(
Univ. Münster
)
An Introduction to the Categorical padic Langlands Program (3/4)
Toby GEE
(
Imperial College
)
Matthew EMERTON
(
Chicago Univ.
)
Eugen HELLMANN
(
Univ. Münster
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
An introduction to the "categorical" approach to the padic Langlands program, in both the "Banach'' and "analytic'' settings.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
Friday, July 22, 2022
9:00 AM
Welcome coffee
Welcome coffee
9:00 AM  9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM
On Moduli Spaces of Local Langlands Parameters (1/2)

JeanFrançois DAT
(
IMJPRG
)
On Moduli Spaces of Local Langlands Parameters (1/2)
JeanFrançois DAT
(
IMJPRG
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
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.
10:40 AM
Coffe break
Coffe break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Explicit Constructions of Automorphic Forms (2/2)

Wee Teck GAN
(
Nat. Univ. Singapour
)
Explicit Constructions of Automorphic Forms (2/2)
Wee Teck GAN
(
Nat. Univ. Singapour
)
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
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.
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
Supercuspidal Representations: Construction, Classification, and Characters (1/2)

Jessica FINTZEN
(
Duke Univ. & Cambridge Univ.
)
Supercuspidal Representations: Construction, Classification, and Characters (1/2)
Jessica FINTZEN
(
Duke Univ. & Cambridge Univ.
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
We have seen in the first week of the summer school that the buildings blocks for irreducible representations of padic 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 padic world and discrete series representations of real algebraic Lie groups. A key ingredient for the construction of supercuspidal representations is the BruhatTits theory and MoyPrasad filtration, which we will introduce in the lecture series.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
The Langlands Program and the Moduli of Bundles on the Curve (2/3)

Laurent FARGUES
(
IMJPRG
)
Peter SCHOLZE
(
Univ. Bonn
)
The Langlands Program and the Moduli of Bundles on the Curve (2/3)
Laurent FARGUES
(
IMJPRG
)
Peter SCHOLZE
(
Univ. Bonn
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
I will speak about my joint work on the geometrization of the local Langlands correspondence.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
Saturday, July 23, 2022
Sunday, July 24, 2022
Monday, July 25, 2022
9:00 AM
Welcome coffee
Welcome coffee
9:00 AM  9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM
An Introduction to the Categorical padic Langlands Program (4/4)

Eugen HELLMANN
(
Univ. Münster
)
Toby GEE
(
Imperial College
)
Matthew EMERTON
(
Chicago Univ.
)
An Introduction to the Categorical padic Langlands Program (4/4)
Eugen HELLMANN
(
Univ. Münster
)
Toby GEE
(
Imperial College
)
Matthew EMERTON
(
Chicago Univ.
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
An introduction to the "categorical" approach to the padic Langlands program, in both the "Banach'' and "analytic'' settings.
10:40 AM
Coffe break
Coffe break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Supercuspidal Representations: Construction, Classification, and Characters (2/2)

Jessica FINTZEN
(
Duke Univ. & Cambridge Univ.
)
Supercuspidal Representations: Construction, Classification, and Characters (2/2)
Jessica FINTZEN
(
Duke Univ. & Cambridge Univ.
)
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
We have seen in the first week of the summer school that the buildings blocks for irreducible representations of padic 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 padic world and discrete series representations of real algebraic Lie groups. A key ingredient for the construction of supercuspidal representations is the BruhatTits theory and MoyPrasad filtration, which we will introduce in the lecture series.
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
Branching Laws: Homological Aspects

Dipendra PRASAD
(
IIT Bombay
)
Branching Laws: Homological Aspects
Dipendra PRASAD
(
IIT Bombay
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
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.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
On Moduli Spaces of Local Langlands Parameters (2/2)

JeanFrançois DAT
(
IMJPRG
)
On Moduli Spaces of Local Langlands Parameters (2/2)
JeanFrançois DAT
(
IMJPRG
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
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.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
Tuesday, July 26, 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 Langlands Program and the Moduli of Bundles on the Curve (3/3)

Peter SCHOLZE
(
Univ. Bonn
)
Laurent FARGUES
(
IMJPRG
)
The Langlands Program and the Moduli of Bundles on the Curve (3/3)
Peter SCHOLZE
(
Univ. Bonn
)
Laurent FARGUES
(
IMJPRG
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
I will speak about my joint work on the geometrization of the local Langlands correspondence.
10:40 AM
Coffee break
Coffee break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Shimura Varieties and Modularity (1/3)

Ana CARAIANI
(
Imperial College
)
Sug Woo SHIN
(
UC Berkeley
)
Shimura Varieties and Modularity (1/3)
Ana CARAIANI
(
Imperial College
)
Sug Woo SHIN
(
UC Berkeley
)
11:00 AM  12:00 PM
Room: Marilyn and James Simons Conference Center
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 HodgeTate period morphism.
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 and Modularity (2/3)

Sug Woo SHIN
(
UC Berkeley
)
Ana CARAIANI
(
Imperial College
)
Shimura Varieties and Modularity (2/3)
Sug Woo SHIN
(
UC Berkeley
)
Ana CARAIANI
(
Imperial College
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
We describe the CalegariGeraghty method for proving modularity lifting theorems beyond the classical setting of the TaylorWiles method. We discuss the three conjectures that this method relies on (existence of Galois representations, localglobal compatibility and vanishing of cohomology outside a certain range of degrees) and their current status, and then explain the commutative algebra underlying the method.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
Geometric and Arithmetic Theta Correspondences (1/2)

Chao LI
(
Columbia Univ.
)
Geometric and Arithmetic Theta Correspondences (1/2)
Chao LI
(
Columbia Univ.
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
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.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
Wednesday, July 27, 2022
9:00 AM
Welcome coffee
Welcome coffee
9:00 AM  9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM
Shimura Varieties and Modularity (3/3)

Ana CARAIANI
(
Imperial College
)
Sug Woo SHIN
(
UC Berkeley
)
Shimura Varieties and Modularity (3/3)
Ana CARAIANI
(
Imperial College
)
Sug Woo SHIN
(
UC Berkeley
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
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 CaraianiScholze and Koshikawa, both of which rely on the geometry of the HodgeTate period morphism for the corresponding Shimura varieties. Finally, we explain how these vanishing results can be applied to localglobal compatibility questions for the Galois representations constructed in the first lecture.
10:40 AM
Coffee break
Coffee break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Geometric and Arithmetic Theta Correspondences (2/2)

Chao LI
(
Columbia Univ.
)
Geometric and Arithmetic Theta Correspondences (2/2)
Chao LI
(
Columbia Univ.
)
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
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.
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
Hamiltonian Actions and Langlands Duality (1/2)

Akshay VENKATESH
(
IAS
)
Hamiltonian Actions and Langlands Duality (1/2)
Akshay VENKATESH
(
IAS
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
I will give a gentle introduction to my joint work with BenZvi 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.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
Hamiltonian Actions and Langlands Duality (2/2)

Akshay VENKATESH
(
IAS
)
Hamiltonian Actions and Langlands Duality (2/2)
Akshay VENKATESH
(
IAS
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
I will give a gentle introduction to my joint work with BenZvi 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.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
Thursday, July 28, 2022
9:00 AM
Welcome coffee
Welcome coffee
9:00 AM  9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM
Highdimensional Gross–Zagier Formula (1/2)

Wei ZHANG
(
MIT
)
Highdimensional Gross–Zagier Formula (1/2)
Wei ZHANG
(
MIT
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
I'll discuss various generalizations of the GrossZagier formula to highdimensional 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.
10:40 AM
Coffee break
Coffee break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Between Coherent and Constructible Local Langlands Correspondences

David BENZVI
(
UT Austin
)
Between Coherent and Constructible Local Langlands Correspondences
David BENZVI
(
UT Austin
)
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
(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 KazhdanLusztig 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 nonarchimedean 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 FarguesFontaine curve, and its relation to constructible local Langlands via circle actions. In the nonarchimedean 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.
12:30 PM
Q&A
Q&A
12:30 PM  1:00 PM
Room: Marilyn and James Simons Conference Center
1:00 PM
Buffet Lunch at IHES
Buffet Lunch at IHES
1:00 PM  3:00 PM
Room: Marilyn and James Simons Conference Center
3:00 PM
Derived Aspects of the Langlands Program (1/3)

Tony FENG
(
MIT
)
Michael HARRIS
(
Columbia Univ.
)
Derived Aspects of the Langlands Program (1/3)
Tony FENG
(
MIT
)
Michael HARRIS
(
Columbia Univ.
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
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.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
Local and Global Questions “Beyond Endoscopy” (1/2)

Yiannis SAKELLARIDIS
(
Johns Hopkins Univ.
)
Local and Global Questions “Beyond Endoscopy” (1/2)
Yiannis SAKELLARIDIS
(
Johns Hopkins Univ.
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
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 BravermanKazhdan, also around 2000, to generalize the GodementJacquet 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.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center
Friday, July 29, 2022
9:00 AM
Welcome coffee
Welcome coffee
9:00 AM  9:20 AM
Room: Marilyn and James Simons Conference Center
9:20 AM
Derived Aspects of the Langlands Program (2/3)

Tony FENG
(
MIT
)
Michael HARRIS
(
Columbia Univ.
)
Derived Aspects of the Langlands Program (2/3)
Tony FENG
(
MIT
)
Michael HARRIS
(
Columbia Univ.
)
9:20 AM  10:40 AM
Room: Marilyn and James Simons Conference Center
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.
10:40 AM
Coffee break
Coffee break
10:40 AM  11:00 AM
Room: Marilyn and James Simons Conference Center
11:00 AM
Highdimensional Gross–Zagier Formula (2/2)

Wei ZHANG
(
MIT
)
Highdimensional Gross–Zagier Formula (2/2)
Wei ZHANG
(
MIT
)
11:00 AM  12:20 PM
Room: Marilyn and James Simons Conference Center
I'll discuss various generalizations of the GrossZagier formula to highdimensional 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.
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
Local and Global Questions “Beyond Endoscopy” (2/2)

Yiannis SAKELLARIDIS
(
Johns Hopkins Univ.
)
Local and Global Questions “Beyond Endoscopy” (2/2)
Yiannis SAKELLARIDIS
(
Johns Hopkins Univ.
)
3:00 PM  4:00 PM
Room: Marilyn and James Simons Conference Center
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 BravermanKazhdan, also around 2000, to generalize the GodementJacquet 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.
4:00 PM
Coffee break
Coffee break
4:00 PM  4:20 PM
Room: Marilyn and James Simons Conference Center
4:20 PM
Derived Aspects of the Langlands Program (3/3)

Tony FENG
(
MIT
)
Michael HARRIS
(
Columbia Univ.
)
Derived Aspects of the Langlands Program (3/3)
Tony FENG
(
MIT
)
Michael HARRIS
(
Columbia Univ.
)
4:20 PM  5:20 PM
Room: Marilyn and James Simons Conference Center
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.
5:20 PM
Q&A
Q&A
5:20 PM  6:00 PM
Room: Marilyn and James Simons Conference Center