The Local Langlands Conjecture (1/3)

• Olivier TAÏBI (ENS Lyon)

Room: Marilyn and James Simons Conference Center We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.

Some Perspectives on Eisenstein Series (1/2)

• Erez LAPID (Weizmann Institute)

Room: Marilyn and James Simons Conference Center This is a review of some developments in the theory of the Eisenstein series since Corvallis.

Q&A Room: Marilyn and James Simons Conference Center

The Local Langlands Conjecture (2/3)

• Olivier TAÏBI (ENS Lyon)

Room: Marilyn and James Simons Conference Center We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.

Shimura Varieties: outline (1/3)

• Sophie MOREL (ENS Lyon)

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).

Q&A Room: Marilyn and James Simons Conference Center

Introduction to the (Relative) Trace Formula (1/2)

• Pierre-Henri CHAUDOUARD (IMJ-PRG)

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

Some Perspectives on Eisenstein Series (2/2)

• Erez LAPID (Weizmann Institute)

Room: Marilyn and James Simons Conference Center This is a review of some developments in the theory of the Eisenstein series since Corvallis.

Q&A Room: Marilyn and James Simons Conference Center

Shimura Varieties: outline (2/3)

• Sophie MOREL (ENS Lyon)

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).

The Local Langlands Conjecture (3/3)

• Olivier TAÏBI (ENS Lyon)

Room: Marilyn and James Simons Conference Center We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.

Q&A Room: Marilyn and James Simons Conference Center

Introduction to the (Relative) Trace Formula (2/2)

• Pierre-Henri CHAUDOUARD (IMJ-PRG)

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

Arthur’s Conjectures and the Orbit Method for Real Reductive Groups

• Lucas MASON-BROWN (Univ. Oxford)

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

Q&A Room: Marilyn and James Simons Conference Center

Introduction to Shtukas and their Moduli (1/3)

• Zhiwei YUN (MIT)

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.

Shimura Varieties: outline (3/3)

• Sophie MOREL (ENS Lyon)

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).

Q&A Room: Marilyn and James Simons Conference Center

The Relative Langlands Program (1/3)

• Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)

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

Coherent Sheaves on the Stack of Langlands Parameters (1/3)

• Xinwen ZHU (Caltech)

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.

Q&A Room: Marilyn and James Simons Conference Center

Introduction to Shtukas and their Moduli (2/3)

• Zhiwei YUN (MIT)

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.

Introduction to Shtukas and their Moduli (3/3)

• Zhiwei YUN (MIT)

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.

Q&A Room: Marilyn and James Simons Conference Center

The Relative Langlands Program (2/3)

• Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)

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

Coherent Sheaves on the Stack of Langlands Parameters (2/3)

• Xinwen ZHU (Caltech)

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.

Q&A Room: Marilyn and James Simons Conference Center

Coherent Sheaves on the Stack of Langlands Parameters (3/3)

• Xinwen ZHU (Caltech)

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.

The Relative Langlands Program (3/3)

• Raphaël BEUZART-PLESSIS (Univ. Aix-Marseille)

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

Q&A Room: Marilyn and James Simons Conference Center

A Brief Introduction to the Trace Formula and its Stabilization (1/2)

• Tasho KALETHA (Univ. Michigan)

Room: Marilyn and James Simons Conference Center 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.

Local Shtukas and the Langlands Program (1/2)

• Jared WEINSTEIN (Boston Univ.)

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 Fargues-Fontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the lectures of Fargues-Scholze.

Q&A Room: Marilyn and James Simons Conference Center

Orbital Integrals, Moduli Spaces and Invariant Theory (1/3)

• Bao Châu NGÔ (Chicago Univ.)

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.

Orbital Integrals, Moduli Spaces and Invariant Theory (2/3)

• Bao Châu NGÔ (Chicago Univ.)

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.

Q&A Room: Marilyn and James Simons Conference Center

Local Shtukas and the Langlands Program (2/2)

• Jared WEINSTEIN (Boston Univ.)

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 Fargues-Fontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the lectures of Fargues-Scholze.

A Brief Introduction to the Trace Formula and its Stabilization (2/2)

• Tasho KALETHA (Univ. Michigan)

Room: Marilyn and James Simons Conference Center 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.

Q&A Room: Marilyn and James Simons Conference Center

Cohomology Sheaves of Stacks of Shtukas (1/2)

• Cong XUE (IMJ-PRG)

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 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.

Cohomology Sheaves of Stacks of Shtukas (2/2)

• Cong XUE (IMJ-PRG)

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 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.

Q&A Room: Marilyn and James Simons Conference Center

An Introduction to the Categorical p-adic Langlands Program (1/4)

• Matthew EMERTON (Chicago Univ.)
• Eugen HELLMANN (Univ. Münster)
• Toby GEE (Imperial College)

Room: Marilyn and James Simons Conference Center An introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach'' and "analytic'' settings.

Orbital Integrals, Moduli Spaces and Invariant Theory (3/3)

• Bao Châu NGÔ

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.

Q&A Room: Marilyn and James Simons Conference Center

What Does Geometric Langlands Mean to a Number Theorist? (1/2)

• Sam RASKIN (Univ. Texas)

Room: Marilyn and James Simons Conference Center

What Does Geometric Langlands Mean to a Number Theorist? (2/2)

• Sam RASKIN (Univ. Texas)

Room: Marilyn and James Simons Conference Center

Q&A Room: Marilyn and James Simons Conference Center

Panel Discussion with Participants of the Corvallis Conference 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. Paris-Saclay), H. Jacquet (Columbia Univ.), J.P. Labesse (Aix-Marseille Université), K. Ribet (Berkeley Univ.), C. Soulé (CNRS-IHES), M.F. Vignéras (IMJ-PRG).

The Langlands Program and the Moduli of Bundles on the Curve (1/3)

• Peter SCHOLZE (Univ. Bonn)
• Laurent FARGUES (IMJ-PRG)

Room: Marilyn and James Simons Conference Center I will speak about my joint work on the geometrization of the local Langlands correspondence.

Explicit Constructions of Automorphic Forms (1/2)

• Wee Teck GAN (Nat. Univ. Singapour)

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.

Q&A Room: Marilyn and James Simons Conference Center

An Introduction to the Categorical p-adic Langlands Program (2/4)

• Toby GEE (Imperial College)
• Eugen HELLMANN (Univ. Münster)
• Matthew EMERTON (Chicago Univ.)

Room: Marilyn and James Simons Conference Center An introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach'' and "analytic'' settings.

An Introduction to the Categorical p-adic Langlands Program (3/4)

• Toby GEE (Imperial College)
• Eugen HELLMANN (Univ. Münster)
• Matthew EMERTON (Chicago Univ.)

Room: Marilyn and James Simons Conference Center An introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach'' and "analytic'' settings.

Q&A Room: Marilyn and James Simons Conference Center

On Moduli Spaces of Local Langlands Parameters (1/2)

• Jean-François DAT (IMJ-PRG)

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.

Explicit Constructions of Automorphic Forms (2/2)

• Wee Teck GAN (Nat. Univ. Singapour)

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.

Q&A Room: Marilyn and James Simons Conference Center

Supercuspidal Representations: Construction, Classification, and Characters (1/2)

• Jessica FINTZEN (Duke Univ. & Cambridge Univ.)

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 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.

The Langlands Program and the Moduli of Bundles on the Curve (2/3)

• Peter SCHOLZE (Univ. Bonn)
• Laurent FARGUES (IMJ-PRG)

Room: Marilyn and James Simons Conference Center I will speak about my joint work on the geometrization of the local Langlands correspondence.

Q&A Room: Marilyn and James Simons Conference Center

An Introduction to the Categorical p-adic Langlands Program (4/4)

• Eugen HELLMANN (Univ. Münster)
• Matthew EMERTON (Chicago Univ.)
• Toby GEE (Imperial College)

Room: Marilyn and James Simons Conference Center An introduction to the "categorical" approach to the p-adic Langlands program, in both the "Banach'' and "analytic'' settings.

Supercuspidal Representations: Construction, Classification, and Characters (2/2)

• Jessica FINTZEN (Duke Univ. & Cambridge Univ.)

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 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.

Q&A Room: Marilyn and James Simons Conference Center

Branching Laws: Homological Aspects

• Dipendra PRASAD (IIT Bombay)

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.

On Moduli Spaces of Local Langlands Parameters (2/2)

• Jean-François DAT (IMJ-PRG)

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.

Q&A Room: Marilyn and James Simons Conference Center

The Langlands Program and the Moduli of Bundles on the Curve (3/3)

• Laurent FARGUES (IMJ-PRG)
• Peter SCHOLZE (Univ. Bonn)

Room: Marilyn and James Simons Conference Center I will speak about my joint work on the geometrization of the local Langlands correspondence.

Shimura Varieties and Modularity (1/3)

• Sug Woo SHIN (UC Berkeley)
• Ana CARAIANI (Imperial College)

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 Hodge-Tate period morphism.

Q&A Room: Marilyn and James Simons Conference Center

Shimura Varieties and Modularity (2/3)

• Ana CARAIANI (Imperial College)
• Sug Woo SHIN (UC Berkeley)

Room: Marilyn and James Simons Conference Center 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.

Geometric and Arithmetic Theta Correspondences (1/2)

• Chao LI (Columbia Univ.)

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.

Q&A Room: Marilyn and James Simons Conference Center

Shimura Varieties and Modularity (3/3)

• Sug Woo SHIN (UC Berkeley)
• Ana CARAIANI (Imperial College)

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 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.

Geometric and Arithmetic Theta Correspondences (2/2)

• Chao LI (Columbia Univ.)

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.

Q&A Room: Marilyn and James Simons Conference Center

Hamiltonian Actions and Langlands Duality (1/2)

• Akshay VENKATESH (IAS)

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

Hamiltonian Actions and Langlands Duality (2/2)

• Akshay VENKATESH (IAS)

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

Q&A Room: Marilyn and James Simons Conference Center

High-dimensional Gross–Zagier Formula (1/2)

• Wei ZHANG (MIT)

Room: Marilyn and James Simons Conference Center 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.

Between Coherent and Constructible Local Langlands Correspondences

• David BEN-ZVI (UT Austin)

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

Q&A Room: Marilyn and James Simons Conference Center

Derived Aspects of the Langlands Program (1/3)

• Tony FENG (MIT)
• Michael HARRIS (Columbia Univ.)

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.

Local and Global Questions “Beyond Endoscopy” (1/2)

• Yiannis SAKELLARIDIS (Johns Hopkins Univ.)

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 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.

Q&A Room: Marilyn and James Simons Conference Center

Derived Aspects of the Langlands Program (2/3)

• Michael HARRIS (Columbia Univ.)
• Tony FENG (MIT)

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.

High-dimensional Gross–Zagier Formula (2/2)

• Wei ZHANG (MIT)

Room: Marilyn and James Simons Conference Center 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.

Q&A Room: Marilyn and James Simons Conference Center

Local and Global Questions “Beyond Endoscopy” (2/2)

• Yiannis SAKELLARIDIS (Johns Hopkins Univ.)

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 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.

Derived Aspects of the Langlands Program (3/3)

• Michael HARRIS (Columbia Univ.)
• Tony FENG (MIT)

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.

Q&A Room: Marilyn and James Simons Conference Center