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

REMINDER: the participants of the Summer School who will come in person have been selected among a lot of applicants. Others, who were not selected but put on a waiting list won't participate in person because the capacity of the conference center has been reached.

Therefore, if you are interested in participating but did not apply or haven't been selected or invited to come, please register to participate through Zoom and don't come to IHES. There are no more seats available and those who did apply and were selected through the application process have the priority to participate in person.

Thank you for your understanding and cooperation!

************

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.)
  • PRASAD Dipendra (IIT Bombay)
  • 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.


 

Registration
Zoom Registration form
Contact: Elisabeth Jasserand
    • 09:00 09:20
      Welcome coffee and registration 20m
    • 09:20 10:40
      The Local Langlands Conjecture (1/3) 1h 20m
      Speaker: Prof. Olivier TAÏBI (ENS Lyon)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      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 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break (buffet at IHES) 2h
    • 15:00 16:00
      The Local Langlands Conjecture (2/3) 1h
      Speaker: Prof. Olivier TAÏBI (ENS Lyon)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      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)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome Coffee 20m
    • 09:20 11:00
      Introduction to the (Relative) Trace Formula (1/2) 1h 40m
      Speaker: Prof. Pierre-Henri CHAUDOUARD (IMJ-PRG)
    • 11:00 11:20
      Coffee break 20m
    • 11:20 12:20
      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 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      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)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      The Local Langlands Conjecture (3/3) 1h
      Speaker: Prof. Olivier TAÏBI (ENS Lyon)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      Introduction to the (Relative) Trace Formula (2/2) 1h 20m
      Speaker: Prof. Pierre-Henri CHAUDOUARD (IMJ-PRG)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      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 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Introduction to Shtukas and their Moduli (1/3) 1h
      Speaker: Prof. Zhiwei YUN (MIT)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      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)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      The Relative Langlands Program (1/3) 1h 20m
      Speaker: Prof. Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      Coherent Sheaves on the Stack of Langlands Parameters (1/3) 1h 20m
      Speaker: Prof. Xinwen ZHU (Caltech)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Introduction to Shtukas and their Moduli (2/3) 1h
      Speaker: Prof. Zhiwei YUN (MIT)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      Introduction to Shtukas and their Moduli (3/3) 1h
      Speaker: Prof. Zhiwei YUN (MIT)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      The Relative Langlands Program (2/3) 1h 20m
      Speaker: Prof. Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      Coherent Sheaves on the Stack of Langlands Parameters (2/3) 1h 20m
      Speaker: Prof. Xinwen ZHU (Caltech)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Coherent Sheaves on the Stack of Langlands Parameters (3/3) 1h
      Speaker: Prof. Xinwen ZHU (Caltech)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      The Relative Langlands Program (3/3) 1h
      Speaker: Prof. Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcom coffee 20m
    • 09:20 10:40
      A Brief Introduction to the Trace Formula and its Stabilization (1/2) 1h 20m
      Speaker: Prof. Tasho KALETHA (Univ. Michigan)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      Local Shtukas and the Langlands Program (1/2) 1h 20m
      Speaker: Prof. Jared WEINSTEIN (Boston Univ.)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break (buffet at IHES) 2h
    • 15:00 16:00
      Orbital integrals, Hitchin type module spaces and invariant theory (title to be confirmed) (1/3) 1h
      Speaker: Prof. Bao Châu NGÔ (Chicago Univ.)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      Orbital integrals, Hitchin type module spaces and invariant theory (title to be confirmed) (2/3) 1h
      Speaker: Prof. Bao Châu NGÔ (Chicago Univ.)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Coffee 20m
    • 09:20 10:40
      Local Shtukas and the Langlands Program (2/2) 1h 20m
      Speaker: Prof. Jared WEINSTEIN (Boston Univ.)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      A Brief Introduction to the Trace Formula and its Stabilization (2/2) 1h 20m
      Speaker: Prof. Tasho KALETHA (Univ. Michigan)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Cohomology Sheaves of Stacks of Shtukas (1/2) 1h
      Speaker: Prof. Cong XUE (IMJ-PRG)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      Cohomology Sheaves of Stacks of Shtukas (2/2) 1h
      Speaker: Prof. Cong XUE (IMJ-PRG)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      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 HELLMAN (Univ. Münster) , Prof. Matthew EMERTON (Chicago Univ.) , Prof. Toby GEE (Imperial College)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      Orbital integrals, Hitchin type module spaces and invariant theory (title to be confirmed) (3/3) 1h 20m
      Speaker: Prof. Bao Châu NGÔ
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Buffet-lunch 2h
    • 15:00 16:00
      What does geometric Langlands mean to a number theorist? (1/2) 1h
      Speaker: Prof. Sam RASKIN (Univ. Texas)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      What does geometric Langlands mean to a number theorist? (2/2) 1h
      Speaker: Prof. Sam RASKIN (Univ. Texas)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      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 11:00
      Coffee break 20m
    • 11:00 12:20
      Explicit Constructions of Automorphic Forms (1/2) 1h 20m
      Speaker: Prof. Wee Teck GAN (Nat. Univ. Singapour)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Shimura Varieties and Modularity (1/3) 1h
      Speakers: Prof. Ana CARAIANI (Imperial College) , Prof. Sug Woo SHIN (UC Berkeley)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      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 HELLMAN (Univ. Münster) , Prof. Matthew EMERTON (Chicago Univ.) , Prof. Toby GEE (Imperial College)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      On Moduli Spaces of Local Langlands Parameters (1/2) 1h 20m
      Speaker: Prof. Jean-François DAT (IMJ-PRG)
    • 10:40 11:00
      Coffe break 20m
    • 11:00 12:20
      Explicit Constructions of Automorphic Forms (2/2) 1h 20m
      Speaker: Prof. Wee Teck GAN (Nat. Univ. Singapour)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Supercuspidal Representations: Construction, Classification, and Characters (1/2) 1h
      Speaker: Prof. Jessica FINTZEN (Duke Univ. & Cambridge Univ.)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      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)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      An Introduction to the Categorical p-adic Langlands Program (3/4) 1h 20m

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

      Speakers: Prof. Eugen HELLMAN (Univ. Münster) , Prof. Matthew EMERTON (Chicago Univ.) , Prof. Toby GEE (Imperial College)
    • 10:40 11:00
      Coffe break 20m
    • 11:00 12:20
      Supercuspidal Representations: Construction, Classification, and Characters (2/2) 1h 20m
      Speaker: Prof. Jessica FINTZEN (Duke Univ. & Cambridge Univ.)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      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)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      On Moduli Spaces of Local Langlands Parameters (2/2) 1h
      Speaker: Prof. Jean-François DAT (IMJ-PRG)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      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 11:00
      Coffee break 20m
    • 11:00 12:20
      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 HELLMAN (Univ. Münster) , Prof. Matthew EMERTON (Chicago Univ.) , Prof. Toby GEE (Imperial College)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Shimura Varieties and Modularity (2/3) 1h
      Speakers: Prof. Ana CARAIANI (Imperial College) , Prof. Sug Woo SHIN (UC Berkeley)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      Geometric and Arithmetic Theta Correspondences (1/2) 1h
      Speaker: Prof. Chao LI (Columbia Univ.)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      Shimura Varieties and Modularity (3/3) 1h 20m
      Speakers: Prof. Ana CARAIANI (Imperial College) , Prof. Sug Woo SHIN (UC Berkeley)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      Geometric and Arithmetic Theta Correspondences (2/2) 1h 20m
      Speaker: Prof. Chao LI (Columbia Univ.)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Hamiltonian Actions and Langlands Duality (1/2) 1h
      Speaker: Prof. Akshay VENKATESH (IAS)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      Hamiltonian Actions and Langlands Duality (2/2) 1h
      Speaker: Prof. Akshay VENKATESH (IAS)
    • 17:20 18:00
      Q&A 40m
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      High-dimensional Gross–Zagier Formula (1/2) 1h 20m
      Speaker: Prof. Wei ZHANG (MIT)
    • 10:40 11:00
      Coffee break 20m
    • 11:00 12:20
      Between Coherent and Constructible Local Langlands Correspondences 1h 20m
      Speaker: Prof. David BEN-ZVI (UT Austin)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      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)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      Local and Global Questions “Beyond Endoscopy” (1/2) 1h
      Speaker: Prof. Yiannis SAKELLARIDIS (Johns Hopkins Univ.)
    • 17:20 18:00
      Q&A 40m
    • 18:00 20:00
      Buffet dinner 2h
    • 09:00 09:20
      Welcome coffee 20m
    • 09:20 10:40
      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 11:00
      Coffee break 20m
    • 11:00 12:20
      High-dimensional Gross–Zagier Formula (2/2) 1h 20m
      Speaker: Prof. Wei ZHANG (MIT)
    • 12:30 13:00
      Q&A 30m
    • 13:00 15:00
      Lunch break 2h
    • 15:00 16:00
      Local and Global Questions “Beyond Endoscopy” (2/2) 1h
      Speaker: Prof. Yiannis SAKELLARIDIS (Johns Hopkins Univ.)
    • 16:00 16:20
      Coffee break 20m
    • 16:20 17:20
      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)
    • 17:20 18:00
      Q&A 40m