Yann LeCun - A Path Towards Autonomous Machine IntelligenceInstitut des Hautes Etudes Scientifiques (IHES)2024-10-15 | Yann LeCun - A Path Towards Autonomous Machine Intelligence04 Laure Saint-Raymond, Professeure Permanente à lIHESInstitut des Hautes Études Scientifiques (IHÉS)2022-11-17 | Exposé "Quelques problèmes mathématiques issus de l'océanographie" de Laure Saint-Raymond, professeure permanente à l'IHES, lors de la cérémonie de clôture de la campagne de levée de fonds de l'IHES le 24 octobre 2022 à la Caisse des Dépôts.Slava Rychkov - Random Field Ising Model and Parisi-Sourlas Supersymmetry (3/4)Institut des Hautes Études Scientifiques (IHÉS)2022-11-16 | Numerical evidence suggests that the Random Field Ising Model loses Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4 and 5 dimensions, while a related model of branched polymers retains these features in any d. I will present a recent theory, developed in 2019-2021 jointly with A. Kaviraj and E. Trevisani and published in [1-4], which aims to explain these facts.
Outline: 1. Random Field Ising Model: phase diagram, well-established facts and experiments. 2. Numerical results for the dimensional reduction of critical exponents: “no” for d=3,4, “yes” for d=5. 3. Parisi-Sourlas supersymmetry implies dimensional reduction 4. Generalities about RG fixed point disappearance 5. Loss of Parisi-Sourlas SUSY via dangerously irrelevant operators? 6. Replica field theory. Cardy field transform “derivation" of Parisi-Sourlas SUSY and its potential loopholes. 7. Replica symmetric interactions in the Cardy basis 8. Leader and follower interactions 9. Classification of leaders 10. Anomalous dimension computations and results. Evidence for the SUSY fixed point instability below ~4.5 11. Future directions and open problems.
Literature: [1] A. Kaviraj, S. Rychkov, E. Trevisani, "Random Field Ising Model and Parisi-Sourlas Supersymmetry I. Supersymmetric CFT," [arXiv:1912.01617] JHEP 2004 (2020) 090 [2] A. Kaviraj, S. Rychkov, E. Trevisani, "Random Field Ising Model and Parisi-Sourlas Supersymmetry II. Renormalization Group”, [arXiv:2009.10087] JHEP 03 (2021) 219 [3] A. Kaviraj, S. Rychkov, E. Trevisani, "The fate of Parisi-Sourlas supersymmetry in Random Field models”, [arXiv:2112.06942] Phys.Rev.Lett. 129 (2022) 045701 [4] A. Kaviraj, E. Trevisani, "Random Field φ^3 Model and Parisi-Sourlas Supersymmetry", [arXiv:2203.12629] JHEP 08 (2022) 290
Slava Rychkov (IHES)03 Marwan Lahoud, Président du Conseil dadministration de lIHESInstitut des Hautes Études Scientifiques (IHÉS)2022-11-15 | Intervention de Marwan Lahoud, Président du Conseil d'administration de l'IHES, lors de la cérémonie de clôture de la campagne de levée de fonds de l'IHES le 24 octobre 2022 à la Caisse des Dépôts.02 Jean Laurent Bonnafé, co-Président du Comité de campagne de lIHESInstitut des Hautes Études Scientifiques (IHÉS)2022-11-15 | Intervention de Jean-Laurent Bonnafé, Administrateur Directeur Général de BNP Paribas et co-Président du Comité de campagne, lors de la cérémonie de clôture de la campagne de levée de fonds de l'IHES le 24 octobre 2022 à la Caisse des Dépôts.05 Eric Lombard, Directeur général de la Caisse des DépôtsInstitut des Hautes Études Scientifiques (IHÉS)2022-11-15 | Intervention d'Eric Lombard, Directeur général de la Caisse des Dépôts, lors de la cérémonie de clôture de la campagne de levée de fonds de l'IHES le 24 octobre 2022 à la Caisse des Dépôts.01 Sophie Quatrehomme, Directrice de la communication du groupe Caisse des DépôtsInstitut des Hautes Études Scientifiques (IHÉS)2022-11-15 | Mot d'accueil de Sophie Quatrehomme, Directrice de la communication du groupe Caisse des Dépôts, lors de la cérémonie de clôture de la campagne de levée de fonds de l'IHES le 24 octobre 2022 à la Caisse des Dépôts.Moments forts de la cérémonie de clôture, 24 octobre 2022Institut des Hautes Études Scientifiques (IHÉS)2022-11-15 | Cérémonie de clôture de la campagne de levée de fonds de l'IHES le 24 octobre 2022 à la Caisse des Dépôts.Slava Rychkov - Random Field Ising Model and Parisi-Sourlas Supersymmetry (2/4)Institut des Hautes Études Scientifiques (IHÉS)2022-11-14 | Numerical evidence suggests that the Random Field Ising Model loses Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4 and 5 dimensions, while a related model of branched polymers retains these features in any d. I will present a recent theory, developed in 2019-2021 jointly with A. Kaviraj and E. Trevisani and published in [1-4], which aims to explain these facts.
Outline: 1. Random Field Ising Model: phase diagram, well-established facts and experiments. 2. Numerical results for the dimensional reduction of critical exponents: “no” for d=3,4, “yes” for d=5. 3. Parisi-Sourlas supersymmetry implies dimensional reduction 4. Generalities about RG fixed point disappearance 5. Loss of Parisi-Sourlas SUSY via dangerously irrelevant operators? 6. Replica field theory. Cardy field transform “derivation" of Parisi-Sourlas SUSY and its potential loopholes. 7. Replica symmetric interactions in the Cardy basis 8. Leader and follower interactions 9. Classification of leaders 10. Anomalous dimension computations and results. Evidence for the SUSY fixed point instability below ~4.5 11. Future directions and open problems.
Literature: [1] A. Kaviraj, S. Rychkov, E. Trevisani, "Random Field Ising Model and Parisi-Sourlas Supersymmetry I. Supersymmetric CFT," [arXiv:1912.01617] JHEP 2004 (2020) 090 [2] A. Kaviraj, S. Rychkov, E. Trevisani, "Random Field Ising Model and Parisi-Sourlas Supersymmetry II. Renormalization Group”, [arXiv:2009.10087] JHEP 03 (2021) 219 [3] A. Kaviraj, S. Rychkov, E. Trevisani, "The fate of Parisi-Sourlas supersymmetry in Random Field models”, [arXiv:2112.06942] Phys.Rev.Lett. 129 (2022) 045701 [4] A. Kaviraj, E. Trevisani, "Random Field φ^3 Model and Parisi-Sourlas Supersymmetry", [arXiv:2203.12629] JHEP 08 (2022) 290
Slava Rychkov (IHES)Slava Rychkov - Random Field Ising Model and Parisi-Sourlas Supersymmetry (1/4)Institut des Hautes Études Scientifiques (IHÉS)2022-11-07 | Numerical evidence suggests that the Random Field Ising Model loses Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4 and 5 dimensions, while a related model of branched polymers retains these features in any d. I will present a recent theory, developed in 2019-2021 jointly with A. Kaviraj and E. Trevisani and published in [1-4], which aims to explain these facts.
Outline: 1. Random Field Ising Model: phase diagram, well-established facts and experiments. 2. Numerical results for the dimensional reduction of critical exponents: “no” for d=3,4, “yes” for d=5. 3. Parisi-Sourlas supersymmetry implies dimensional reduction 4. Generalities about RG fixed point disappearance 5. Loss of Parisi-Sourlas SUSY via dangerously irrelevant operators? 6. Replica field theory. Cardy field transform “derivation" of Parisi-Sourlas SUSY and its potential loopholes. 7. Replica symmetric interactions in the Cardy basis 8. Leader and follower interactions 9. Classification of leaders 10. Anomalous dimension computations and results. Evidence for the SUSY fixed point instability below ~4.5 11. Future directions and open problems.
Literature: [1] A. Kaviraj, S. Rychkov, E. Trevisani, "Random Field Ising Model and Parisi-Sourlas Supersymmetry I. Supersymmetric CFT," [arXiv:1912.01617] JHEP 2004 (2020) 090 [2] A. Kaviraj, S. Rychkov, E. Trevisani, "Random Field Ising Model and Parisi-Sourlas Supersymmetry II. Renormalization Group”, [arXiv:2009.10087] JHEP 03 (2021) 219 [3] A. Kaviraj, S. Rychkov, E. Trevisani, "The fate of Parisi-Sourlas supersymmetry in Random Field models”, [arXiv:2112.06942] Phys.Rev.Lett. 129 (2022) 045701 [4] A. Kaviraj, E. Trevisani, "Random Field φ^3 Model and Parisi-Sourlas Supersymmetry", [arXiv:2203.12629] JHEP 08 (2022) 290
Slava Rychkov (IHES)Pengzi Miao - Recent inequalities on the mass-to-capacity ratioInstitut des Hautes Études Scientifiques (IHÉS)2022-11-03 | On an asymptotically flat 3-manifold, both the mass and the capacity have unit of length, and hence their ratio is a dimensionless quantity. In this talk, I will discuss recent work on establishing new inequalities for the mass-to-capacity ratio on manifolds with nonnegative scalar curvature. Besides revealing additional proofs of the positive mass theorem, applications of these inequalities include new sufficient conditions guaranteeing positive mass via $C^0$-geometry of regions separating the boundary and the infinity. If time permits, a proposal to study manifolds with the mass-to-capacity ratio bounded by one will also be discussed.
Pengzi Miao (University of Miami)Jeff Jauregui - Capacity in low regularity, with connections to general relativityInstitut des Hautes Études Scientifiques (IHÉS)2022-11-03 | The classical concept of capacity generalizes from Euclidean space to complete Riemannian manifolds, and even to suitable classes of metric spaces. I will discuss recent joint work with Raquel Perales and Jim Portegies on understanding capacity in local integral current spaces, describing the behavior of capacity when the background spaces converge in the pointed Sormani--Wenger intrinsic flat sense. Connections between the main results and the concept of total mass in general relativity will be discussed.
Jeff Jauregui (Union College)Simon Townsend - Language Origins: an Animal Communication PerspectiveInstitut des Hautes Études Scientifiques (IHÉS)2022-08-04 | Human language is arguably the most complex communication system currently known, however, the origins of language remain surprisingly elusive. In this lecture, I will revisit this conundrum and illustrate how studying animal communication can provide a much-needed window into the evolutionary roots of language. I will specifically focus on what comparative data exists for syntax in animal vocal systems (i.e. the ability to combine meaningful vocalizations together into larger meaningful structures). Data from primates and birds suggests that this core property of language is not unique to humans but exists, in more simple forms, in animals. I will discuss the evolutionary implications of these findings for reconstructing the emergence of syntax and language more generally.
Simon Townsend (University of Zurich)Shiing-Shen Chern - If Possible Do NothingInstitut des Hautes Études Scientifiques (IHÉS)2022-08-01 | An interview of Shiing Shen Chern by Jean Pierre Bourguignon
Shiing-Shen ChernMichael Harris - 3/3 Derived Aspects of the Langlands ProgramInstitut des Hautes Études Scientifiques (IHÉS)2022-07-29 | 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.
Michael Harris (Columbia Univ.) Tony Feng (MIT)Yiannis Sakellaridis - 2/2 Local and Global Questions “Beyond Endoscopy”Institut des Hautes Études Scientifiques (IHÉS)2022-07-29 | 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 functoriality 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.
Yiannis Sakellaridis (Johns Hopkins Univ.)Michael Harris, Tony Feng - 2/3 Derived Aspects of the Langlands ProgramInstitut des Hautes Études Scientifiques (IHÉS)2022-07-29 | 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.
Michael Harris (Columbia Univ.) Tony Feng (MIT)Wei Zhang - 2/2 High-dimensional Gross–Zagier FormulaInstitut des Hautes Études Scientifiques (IHÉS)2022-07-29 | 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.
Wei Zhang (MIT)Interview with Jessica Fintzen (University of Bonn)Institut des Hautes Études Scientifiques (IHÉS)2022-07-29 | Jessica Fintzen is a professor at the University of Bonn and was an invited speaker at the 2022 Summer School on the Langlands Program at IHES. In this interview, she explains how she became interested in the Langlands Program, the importance of this Summer School, and what she enjoyed the most of these three weeks of talks and discussions. She also shares aspects of her experience as a researcher in mathematics.Yiannis Sakellaridis - 1/2 Local and Global Questions “Beyond Endoscopy”Institut des Hautes Études Scientifiques (IHÉS)2022-07-29 | 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 functoriality 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.
Yiannis Sakellaridis (Johns Hopkins Univ.)Tony Feng - 1/3 Derived Aspects of the Langlands ProgramInstitut des Hautes Études Scientifiques (IHÉS)2022-07-29 | 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.
Michael Harris (Columbia Univ.) Tony Feng (MIT)David Ben-Zvi - Between Coherent and Constructible Local Langlands CorrespondencesInstitut des Hautes Études Scientifiques (IHÉS)2022-07-28 | (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 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.
David Ben-Zvi (UT Austin)Wei Zhang - 1/2 High-dimensional Gross–Zagier FormulaInstitut des Hautes Études Scientifiques (IHÉS)2022-07-28 | 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.
Wei Zhang (MIT)Chao Li - 2/2 Geometric and Arithmetic Theta CorrespondencesInstitut des Hautes Études Scientifiques (IHÉS)2022-07-28 | 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.
Chao Li (Columbia Univ.)Ana Caraiani - 3/3 Shimura Varieties and ModularityInstitut des Hautes Études Scientifiques (IHÉS)2022-07-27 | 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.
Ana Caraiani (Imperial College) Sug Woo Shin (UC Berkeley)Chao Li - 1/2 Geometric and Arithmetic Theta CorrespondencesInstitut des Hautes Études Scientifiques (IHÉS)2022-07-27 | 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.
Chao Li (Columbia Univ.)Ana Caraiani - 2/3 Shimura Varieties and ModularityInstitut des Hautes Études Scientifiques (IHÉS)2022-07-27 | 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.
Ana Caraiani (Imperial College) Sug Woo Shin (UC Berkeley)Laurent Fargues - 3/3 The Langlands Program and the Moduli of Bundles on the Curve (3/3)Institut des Hautes Études Scientifiques (IHÉS)2022-07-27 | I will speak about my joint work about the geometrization of the local Langlands correspondence.
Peter Scholze (Univ. Bonn) Laurent Fargues (IMJ-PRG)Sug Woo Shin - 1/3 Shimura Varieties and ModularityInstitut des Hautes Études Scientifiques (IHÉS)2022-07-26 | 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.
Ana Caraiani (Imperial College) Sug Woo Shin (UC Berkeley)Jean-François Dat - 2/2 On Moduli Spaces of Local Langlands ParametersInstitut des Hautes Études Scientifiques (IHÉS)2022-07-26 | 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.
Jean-François Dat (IMJ-PRG)Dipendra Prasad - Branching laws: homological aspectsInstitut des Hautes Études Scientifiques (IHÉS)2022-07-26 | By this time in the summer school, the audience will have seen the question about decomposing a representation of a group when restricted to a subgroup which is 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 going up to several results which have recently been proved.
Dipendra Prasad (IIT Bombay)Jessica Fintzen - 2/2 Supercuspidal Representations: Construction, Classification, and CharactersInstitut des Hautes Études Scientifiques (IHÉS)2022-07-25 | 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.
Jessica Fintzen (Duke Univ. and Cambridge Univ.)Matthew Emerton - 4/4 An Introduction to the Categorical p-adic Langlands (...)Institut des Hautes Études Scientifiques (IHÉS)2022-07-25 | An introduction to the ``categorical'' approach to the p-adic Langlands program, in both the ``Banach'' and ``analytic'' settings.
Matthew Emerton (Chicago Univ.) Toby Gee (Imperial College) Eugen Hellman (Univ. Münster)Yann LeCun - The Present and Future of Artificial IntelligenceInstitut des Hautes Études Scientifiques (IHÉS)2022-07-25 | Yann LeCun (Meta AI & New York University)Peter Scholze - 2/3 The Langlands Program and the Moduli of Bundles on the CurveInstitut des Hautes Études Scientifiques (IHÉS)2022-07-24 | I will speak about my joint work about the geometrization of the local Langlands correspondence.
Peter Scholze (Univ. Bonn) Laurent Fargues (IMJ-PRG)Jessica Fintzen - 1/2 Supercuspidal Representations: Construction, Classification, and CharactersInstitut des Hautes Études Scientifiques (IHÉS)2022-07-24 | 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.
Jessica Fintzen (Duke Univ. and Cambridge Univ.)Wee Teck Gan - 2/2 Explicit Constructions of Automorphic FormsInstitut des Hautes Études Scientifiques (IHÉS)2022-07-22 | 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.
Wee Teck Gan (Nat. Univ. Singapour)Jean-François Dat - 1/2 On Moduli Spaces of Local Langlands ParametersInstitut des Hautes Études Scientifiques (IHÉS)2022-07-22 | 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.
Jean-François Dat (IMJ-PRG)Matthew Emerton - 3/4 An Introduction to the Categorical p-adic Langlands (...)Institut des Hautes Études Scientifiques (IHÉS)2022-07-22 | An introduction to the ``categorical'' approach to the p-adic Langlands program, in both the ``Banach'' and ``analytic'' settings.
Matthew Emerton (Chicago Univ.) Toby Gee (Imperial College) Eugen Hellman (Univ. Münster)Toby Gee - 2/4 An Introduction to the Categorical p-adic Langlands (...)Institut des Hautes Études Scientifiques (IHÉS)2022-07-21 | An introduction to the ``categorical'' approach to the p-adic Langlands program, in both the ``Banach'' and ``analytic'' settings.
Matthew Emerton (Chicago Univ.) Toby Gee (Imperial College) Eugen Hellman (Univ. Münster)Wee Teck Gan - 1/2 Explicit Constructions of Automorphic FormsInstitut des Hautes Études Scientifiques (IHÉS)2022-07-21 | 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.
Wee Teck Gan (Nat. Univ. Singapour)Peter Scholze - 1/3 The Langlands Program and the Moduli of Bundles on the CurveInstitut des Hautes Études Scientifiques (IHÉS)2022-07-21 | I will speak about my joint work about the geometrization of the local Langlands correspondence.
Peter Scholze (Univ. Bonn) Laurent Fargues (IMJ-PRG)Sam Raskin - 2/2 What does geometric Langlands mean to a number theorist?Institut des Hautes Études Scientifiques (IHÉS)2022-07-21 | Sam Raskin (Univ. Texas)Sam Raskin - 1/2 What does geometric Langlands mean to a number theorist?Institut des Hautes Études Scientifiques (IHÉS)2022-07-21 | Sam Raskin (Univ. Texas)Q2/2: What is your research about?Institut des Hautes Études Scientifiques (IHÉS)2022-07-21 | Hugo Duminil-Copin, a French mathematician specializing in probability, permanent professor at Institut des Hautes Études Scientifiques (IHES) since 2016, explains the mathematical model called Ising model, in the framework of ferromagnetism, and his second research area, percolation.Q2/2: Sur quoi porte votre recherche ?Institut des Hautes Études Scientifiques (IHÉS)2022-07-21 | Hugo Duminil-Copin, mathématicien français spécialisé dans les probabilités, professeur permanent à l’Institut des Hautes Études Scientifiques (IHES) depuis 2016, explique le modèle mathématique appelé modèle d'Ising, dans le cadre du ferromagnétisme, et son deuxième domaine de recherche, la percolation.Bao Chau Ngo - 3/3 Orbital integrals, moduli spaces and invariant theoryInstitut des Hautes Études Scientifiques (IHÉS)2022-07-20 | 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.
Bao Chau Ngo (Chicago Univ.)Eugen Hellman - 1/4 An Introduction to the Categorical p-adic Langlands (...)Institut des Hautes Études Scientifiques (IHÉS)2022-07-20 | An introduction to the ``categorical'' approach to the p-adic Langlands program, in both the ``Banach'' and ``analytic'' settings.
Matthew Emerton (Chicago Univ.) Toby Gee (Imperial College) Eugen Hellman (Univ. Münster)Cong Xue - 2/2 Cohomology Sheaves of Stacks of ShtukasInstitut des Hautes Études Scientifiques (IHÉS)2022-07-20 | 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.
Cong Xue (IMJ-PRG)Cong Xue - 1/2 Cohomology Sheaves of Stacks of ShtukasInstitut des Hautes Études Scientifiques (IHÉS)2022-07-19 | 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.
Cong Xue (IMJ-PRG)Tasho Kaletha - 2/2 A Brief Introduction to the Trace Formula and its StabilizationInstitut des Hautes Études Scientifiques (IHÉS)2022-07-19 | 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.