Representation Theory and Noncommutative Geometry, Paris

6 janv. 2025, 08:30 → 4 avr. 2025, 18:30 Europe/Paris

Amphitheater Darboux (IHP)

 

Thematic trimester programme with short courses, seminars and workshops

January 6 to April 4, 2025 at the Institut Henri Poincaré, Paris

REGISTRATION IS FREE BUT MANDATORY

 

The trimester is part of an ongoing effort to bridge two fields of Mathematics : the representation theory of locally compact groups and the theory of operator algebras. These research domains share origins in harmonic analysis, spectral theory and quantum mechanics but grew in separate directions. Recent progress in representation theory, involving especially non-Riemannian symmetric spaces and spherical varieties, and new tools developed in operator algebras, especially those involving K-theory and the other methods of non-commutative geometry, offer exciting prospects for new work at the interface between the two fields. 

Research workshops  and mini-courses will be the occasion for researchers at every career stage to get acquainted with recent developments in these areas, exchange ideas and explore new research directions. 

Graduate students and post-doctoral researchers whose work connects to one or more of the topics of the trimester are particularly encouraged to participate.

Timeline for the trimester:

January 6 to 10 — Introductory school at CIRM (Marseille): Methods in representation theory and operator algebras

January 13 to 17 — Minicourses: branching problems and symmetry-breaking

Coordinated by Toshiyuki Kobayashi, Michaël Pevzner and Birgit Speh.

• Introduction to representation theory, by Birgit Speh (3h)
• Branching problems in representation theory, by Toshiyuki Kobayashi (3h)
• Different aspects of Rankin–Cohen operators, by Michael Pevzner (2h)
• Geometric and analytic aspects of branching laws, by Bent Ørsted (2h)
• Lectures by Toshihisa Kubo and Hideko Sekiguchi

January 20 to 24 — Workshop 1: Intertwining operators and geometry

January 27 to 31 and February 3 to 7 — Minicourses: the Baum–Connes conjecture and representation theory

• Introduction to the Baum-Connes conjecture, by Alain Valette (10h)
• The Baum-Connes conjecture for real reductive groups, by Nigel Higson (10h)
• Lecture by Pierre Julg

February 5 — Topic day: Operator algebras, index theory and geometry

Coordinated by Pierre Clare and Nicolas Prudhon in celebration of Pierre Julg's career

February 10 to 14 — Research talks on NCG, survey lecture and Carte Blanche

• Talks by early career researchers in noncommutative geometry (program)
• Lecture: Symmetry breaking under translations, by Toshiyuki Kobayashi (joint with AIM-RTNCG online)
• Conférence Carte Blanche: «Quand la symétrie prédit», by Alexandre Afgoustidis at Maison Poincaré (in French)

February 17 to 21 — Mini-courses and research lectures

• Mini-course: Structure of tempered homogeneous spaces, by Toshiyuki Kobayashi

• Lecture: The Dolbeault-Dirac operator on the irreducible quantum flag manifolds, by Fredy Diaz-Garcia
• Lecture: Parthasarathy formulas and beyond, by Nicolas Prudhon (joint with AIM-RTNCG online)
• Mini-course: Twisted Ruelle zeta function, complex-valued analytic torsion and the Fried's conjecture, by Polyxeni Spilioti

 

February 24 to 28 — Workshop 2: Tempered representations and K-theory

February 27 — Public lecture by Alain Connes (IHES)

Topos and Noncommutative Geometry : Two Perspectives on Space and Numbers

At 16:30 in Amphithéâtre Hermite. Résumé / Abstract

March 3 to 7 — p-adic week

Coordinated by Anne-Marie Aubert, Hung Yean Loke and Monica Nevins. 

• Mini-course: The Bernstein center, by David Renard (4h30)
• Mini-course: Hochschild homology of reductive p-adic groups, by Maarten Solleveld (4h)
• Research talks by R. Abdellatif, D. Brajkovic Zoric, P. Cui, M. Emory, J. Fintzen, G. Henniart, R. Plymen, V. Sécherre, D. Schwein, A. Szumovicz

March 10 to 14 — Theta correspondence week

Coordinated by Hung Yean Loke and Tomasz Przebinda.

• Mini-course: The theta correspondence for classical groups over finite fields of odd characteristic, by Anne-Marie Aubert (4h30)
• Mini-course: Minimal representations and applications, by Gordan Savin (4h30)
• Mini-course: Local theta correspondence, by Chengbo Zhu (4h30)
• Research talks by A.-M. Aubert, C. Chen, D. Jiang, J. Ma, A. Merino, A. Pasquale, A. Paull, D. Schwein, H. Sengün, B. Sun, J. Trias

March 17 to 21 — Minicourse: AQFT and causal homogeneous spaces

• Lectures by K.-H. Neeb (short notes for the course, and full version)
• Problem sessions by T. Simon (see schedule) and problem sheets: 
  1. Nets of operator algebras and AQFT and solution
  2. Euler elements and causal homogeneous spaces and solution
  3. Crown domains and analytic continuation and solution
• Research talk by G. Ólafsson

 

March 24 to 28 — Workshop 3: Analysis on homogeneous spaces and operator algebras

March 27 (16:30) — Lecture by Jeffrey Adams (U. Maryland): the Unitary Dual

March 31 to April 4 — Wrap-up and discussion sessions

Organising committee:

• Alexandre Afgoustidis (CNRS)
• Anne-Marie Aubert (CNRS)
• Pierre Clare (College of William & Mary)
• Jan Frahm (Aarhus University)
• Angela Pasquale (Université de Lorraine) 
• Haluk Şengün (University of Sheffield) 

Scientific committee:

• Nigel Higson (Penn State University)
• Roger Howe (Texas A&M University)
• Toshiyuki Kobayashi (University of Tokyo) 
• Georges Skandalis (IMJ PRG)
• Birgit Speh (Cornell University)

Back to the thematic trimester main page 

 

------------------------------------

Programme coordinated by the Centre Emile Borel at IHP

Sorbonne Université - CNRS

 

Feb11-12_prog.pdf

JulgFest_prog.pdf

MC_Tempered_spaces_prog.pdf

Poster-T1-2025 RTNCG.pdf

Public_lecture_Connes_ToposNCG_va.pdf

Public_lecture_Connes_ToposNCG_vf.pdf

CIRM school notes and slides

• Crisp_CIRM_slides.pdf
• Notes from T. Crisp's mini-course
• Slides of T. Kobayashi's survey lecture
• VanDenBan_CIRM_Lecture1.pdf
• VanDenBan_CIRM_Lecture2.pdf
• VanDenBan_CIRM_Lecture3.pdf
• VanDenBan_CIRM_Lecture4.pdf
• VanDenBan_CIRM_Lecture5.pdf

Mini-course: AQFT and causal homogeneous spaces

• AQFT_causal_short_notes.pdf
• Exercise_Sheet_1_AQFT.pdf
• Exercise_Sheet_2_AQFT.pdf
• Exercise_Sheet_3_AQFT.pdf
• MC_AQFT_prog.pdf
• Solution_Sheet_1_AQFT.pdf
• Solution_Sheet_2_AQFT.pdf
• Solution_Sheet_3_AQFT.pdf

Mini-course: Structure of tempered homogeneous spaces

• I. Dynamical Approach (February 17, 2025)
• II. Combinatorics Approach (February 20, 2025)
• III. Limit Algebras (February 23, 2025)

Mini-course: The theta correspondence for classical groups over finite fields of odd characteristic

• Aubert_Theta_I.pdf
• Aubert_Theta_II.pdf
• Aubert_Theta_III.pdf

Mini-courses: branching and symmetry-breaking

• Kubo_Tutorial_Worksheet.pdf
• Minicourse on branching (T. Kobayashi) - Lecture 1/3
• Minicourse on branching (T. Kobayashi) - Lecture 2/3
• Minicourse on branching (T. Kobayashi) - Lecture 3/3
• Pevzner_RC_Lecture_1.pdf
• Pevzner_RC_Lecture_2.pdf

Workshop 1: Intertwining operators and geometry Slides_250120_Kobayashi

Workshop 2: Tempered representations and K-theory Slides_250227_Aubert.pdf

Workshop 3: Analysis on homogeneous spaces and operator algebras 2025IHP_Sakellaridis.pdf

lun. 13 janvier

11:00 → 12:00 Introduction to Representation Theory, 1/3

The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading.

Lecture 1. We will introduce discuss the basic important concepts: Admissible representations, differentiable vectors, equivalence of representations, irreducible representations, unitary representations, (g, K)-modules. Time permitting we will also introduce Cartan subgroups, the Iwasawa decomposition and parabolic subgroups.

Orateur: Birgit Speh (Cornell University)

14:00 → 15:00 Branching Problems in Representation Theory, 1/3

The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches.
Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods.
In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible.
A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Orateur: Prof. Toshiyuki Kobayashi (The University of Tokyo)

15:30 → 16:30 Tutorial

Président de session: Toshihisa Kubo (Ryukoku University)

mar. 14 janvier

11:00 → 12:00 Different Aspects of Rankin–Cohen Operators, 1/2 (online)

Rankin-Cohen brackets provide a fundamental example of “non-elementary” differential symmetry breaking operators. We will explore the combinatorial structure of these operators through multiple perspectives and discuss several approaches to find their higher dimensions analogues within the representation-theoretical perspective of branching problems.

Orateur: Michael Pevzner

14:00 → 15:00 Introduction to Representation Theory, 2/3

The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading.

Lecture 2. We will discuss the representations of SU(2), SL(2, R) and of SL(2, C).

Orateur: Birgit Speh (Cornell University)

15:30 → 16:30 Tutorial

Président de session: Toshihisa Kubo (Ryukoku University)

mer. 15 janvier

10:30 → 12:00 Different Aspects of Rankin–Cohen Operators, 2/2 (online)

Rankin-Cohen brackets provide a fundamental example of “non-elementary” differential symmetry breaking operators. We will explore the combinatorial structure of these operators through multiple perspectives and discuss several approaches to find their higher dimensions analogues within the representation-theoretical perspective of branching problems.

Orateur: Michael Pevzner

14:00 → 15:00 Branching Problems in Representation Theory, 2/3

The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches.
Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods.
In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible.
A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Orateur: Prof. Toshiyuki Kobayashi (The University of Tokyo)

15:30 → 16:30 Differential Symmetry Breaking Operators and F-method

Let X be a smooth manifold and Y a smooth submanifold of X. Take G'⊂G to be a pair of Lie groups that act on Y⊂X, respectively. We call a differential operator D between the space of smooth sections for a G-equivariant vector bundle over X and that for a G'-equivariant vector bundle over Y a differential symmetry breaking operator (differential SBO for short) if D is G'-intertwining. The F-method is a powerful machinery on the classification and construction of differential SBOs. It allows one to classify and construct differential SBOs by solving a certain system of partial differential equations. The key idea is the Fourier transform of the generalized Verma modules. In this talk we shall overview the F-method in an elementary manner.

Orateur: Toshihisa Kubo (Ryukoku University)

jeu. 16 janvier

10:00 → 11:00 Branching Laws and a Generalized Penrose Transform (online)

I plan to discuss a generalized Radon-Penrose transform and provide examples of explicit branching laws from the perspective of differential equations.

Orateur: Hideko Sekiguchi (University of Tokyo)

11:00 → 12:00 Introduction to Representation Theory, 3/3

The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading.

Lecture 3. We will discuss principal series representations, intertwining operators and the Langlands classification of irreducible representations.

Orateur: Birgit Speh (Cornell University)

15:30 → 16:30 Geometric and Analytic Aspects of Branching Laws, 1/2

In these lectures will be explained some of the basic mathematical theory of group representations, mostly of reductive Lie groups, and their structure via the theory of branching problems and intertwining operators.

Orateur: Bent Ørsted (Aarhus University, Denmark)

Orsted_summary.pdf

ven. 17 janvier

11:00 → 12:00 Geometric and Analytic Aspects of Branching Laws, 2/2

In these lectures will be explained some of the basic mathematical theory of group representations, mostly of reductive Lie groups, and their structure via the theory of branching problems and intertwining operators.

Orateur: Bent Ørsted (Aarhus University, Denmark)

Orsted_summary.pdf

14:00 → 15:00 Branching Problems in Representation Theory, 3/3

The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches.
Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods.
In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible.
A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Orateur: Prof. Toshiyuki Kobayashi (The University of Tokyo)

lun. 20 janvier

09:20 → 17:20 Workshop: Intertwining operators and geometry

See: https://indico.math.cnrs.fr/event/10856/timetable

mar. 21 janvier

09:30 → 18:00 Workshop: Intertwining operators and geometry

See: https://indico.math.cnrs.fr/event/10856/timetable

mer. 22 janvier

09:00 → 12:30 Workshop: Intertwining operators and geometry

See: https://indico.math.cnrs.fr/event/10856/timetable

jeu. 23 janvier

09:30 → 18:00 Workshop: Intertwining operators and geometry

See: https://indico.math.cnrs.fr/event/10856/timetable

ven. 24 janvier

09:00 → 12:30 Workshop: Intertwining operators and geometry

See: https://indico.math.cnrs.fr/event/10856/timetable

lun. 27 janvier

14:00 → 15:00 The Baum-Connes conjecture for real reductive groups 1/10

K-theory for C*-algebras

A second review of K-theory for C-algebras: Bott periodicity, stability and long-exact sequences. Really basic C-algebras: characterization and K-theory. Continuous fields of C-algebras, and the K-theory groups of a continuous field. Mapping cone continuous fields. Open subsets of the spectrum of a C-algebra and ideals. Examples related to SL(2,R).

Orateur: Nigel Higson (Penn State University)

15:00 → 16:00 The Baum-Connes conjecture for real reductive groups 2/10

The Cartan motion group and the Mackey deformation family

The Cartan motion group. The unitary dual and the C-algebra of the motion group. Structure in the unitary dual from the restricted root system. A filtration of the group C-algebra by ideals. The deformation construction associated to a Lie group and a maximal compact subgroup. The associated continuous field of group C-algebras. The Connes-Kasparov morphism in C-algebra K-theory. Relation to the Baum-Connes assembly map. The example of SL(2,R).

Orateur: Nigel Higson (Penn State University)

mar. 28 janvier

14:00 → 15:00 Introduction to the Baum-Connes conjecture 1/10

The Baum-Connes conjecture in a nutshell

A corollary: the Kaplansky-Kadison conjecture on idempotents

Orateur: Alain Valette (Université de Neuchâtel)

15:00 → 16:00 Introduction to the Baum-Connes conjecture 2/10

K-theory for C*-algebras, again

K_0, K_1, stability, Bott periodicity, 6-terms exact sequence

Orateur: Alain Valette (Université de Neuchâtel)

16:00 → 17:00 Cuntz’ K-theoretic amenability revisited

In the early 1980’s, the question was raised of comparing the K-theory groups of the full and reduced C*-algebras of a group $G$. J.Cuntz has described a condition (K-amenability) implying that they are isomorphic. Note that K-amenability is incompatible with Kazhdan’s property T, and is implied by the Haagerup property, a strong negation of property T. In this talk we shall explain that Cuntz’ condition relies on the construction of a $G$-Fredhom module. We shall give such a module in the example of $\mathrm{SL}_2$ on the fields of $p$-adics (Julg-Valette), of complex numbers (Kasparov) and of real numbers (Fox-Haskel and Julg-Kasparov).

Orateur: Pierre Julg (Université d'Orléans)

mer. 29 janvier

14:00 → 15:00 Introduction to the Baum-Connes conjecture 3/10

Kasparov theory

Hilbert C*-modules, operators, compact operators; KK-cycles

Orateur: Alain Valette (Université de Neuchâtel)

15:00 → 16:00 Introduction to the Baum-Connes conjecture 4/10

Equivariant Kasparov theory

KK_G-cycles, the ring R(G), the descent map

Orateur: Alain Valette (Université de Neuchâtel)

16:20 → 17:00 Index theory for manifolds with corners

In this talk we define the boundary index of a singular manifold in a general C*-algebraic framework following the Lescure and Carrillo Rouse approach. Then we explain how it encodes geometrical obstruction for symbols to be associated to nicer operators (Fredholm operators in the classical case). Eventually we extend Lescure and Carrillo Rouse's strategies to compute the associated K-theory groups in the case of families of manifolds with corners using iterated blow-ups.

Orateur: Florian Thiry (Institut de mathématiques de Toulouse)

jeu. 30 janvier

10:00 → 11:00 Introduction to the Baum-Connes conjecture 5/10

Proper actions and the classifying space for proper actions

Orateur: Alain Valette (Université de Neuchâtel)

11:00 → 12:00 Introduction to the Baum-Connes conjecture 6/10

The assembly map and the BC conjecture (with and without coefficients)

Orateur: Alain Valette (Université de Neuchâtel)

ven. 31 janvier

14:00 → 15:00 The Baum-Connes conjecture for real reductive groups 3/10

The imaginary part of the infinitesimal character

Definition of the infinitesimal character. The infinitesimal character of a parabolically induced representation. Continuity of the infinitesimal character. The imaginary part of the infinitesimal character. A filtration of the reduced C-algebra of a real reductive group by ideals. Some low-real-rank examples. A theorem of Bruhat and Harish-Chandra about irreducible parabolic induction. Characterization of the fibers of the imaginary part of the infinitesimal character using tempiric representations. Discrete series representations versus tempiric representations. Characterization of the subquotients for the filtration of the reduced group C-algebra by ideals. The Mackey bijection.

Orateur: Nigel Higson (Penn State University)

15:00 → 16:00 The Baum-Connes conjecture for real reductive groups 4/10

More on the discrete series

Harish-Chandra’s Schwartz space. Tempered representations. The Harish-Chandra-Langlands principle. Asymptotic expansions of matrix coefficients and idempotents in the reduced group C-algebra. Speculations about a C-algebraic approach to the discrete series and the Harish-Chandra principle.

Orateur: Nigel Higson (Penn State University)

lun. 3 février

10:00 → 11:00 The Baum-Connes conjecture for real reductive groups 5/10

Proof of the Bruhat-Harish-Chandra theorem

Distributions from intertwiners between parabolically induced representations. Bruhat decomposition. Bounding the dimensions of the spaces of equivariant distributions. Proof of the theorem.

Orateur: Nigel Higson (Penn State University)

11:00 → 12:00 The Baum-Connes conjecture for real reductive groups 6/10

The Connes-Kasparov isomorphism from the perspective of representation theory

The Mackey deformation family and continuity of the rescaled imaginary part of the infinitesimal character. A filtration of the deformation continuous field by ideals. Rescaling automorphisms, embedding morphisms, and computation of the subquotient continuous fields. Reformulation of the Connes-Kasparov isomorphism. Vogan’s theorem.

Orateur: Nigel Higson (Penn State University)

14:00 → 15:00 Introduction to the Baum-Connes conjecture 7/10

Status of the BC conjecture

Orateur: Alain Valette (Université de Neuchâtel)

15:00 → 16:00 Introduction to the Baum-Connes conjecture 8/10

BC vs CK

Orateur: Alain Valette (Université de Neuchâtel)

mar. 4 février

10:00 → 11:00 Introduction to the Baum-Connes conjecture 9/10

The "Dirac-dual Dirac" method and the gamma-element

Orateur: Alain Valette (Université de Neuchâtel)

11:00 → 12:00 Introduction to the Baum-Connes conjecture 10/10

Nishikawa's recent approach

Orateur: Alain Valette (Université de Neuchâtel)

mer. 5 février

08:45 → 09:40 Admissibility of irreducible representations

This is an account of joint work with Francesca Astengo (Genova) and Bianca Di Blasio (Milano Bicocca).

Harish-Chandra showed that irreducible unitary representations of semisimple Lie groups are admissible; these are easier to understand than general representations. But, as Pierre Julg and others have shown, there are good reasons to look at representations that are not unitary, so it is natural to ask whether ``all'' irreducible representations on (say) Banach spaces are admissible. We show that the answer to this question is closely related to a classical question in functional analysis, the invariant subspace problem.

https://sites.google.com/view/julgfest

Orateur: Michael Cowling (University of New South Wales)

10:15 → 11:10 Groupoids as cyclic sets

Associated to a groupoid is a natural cyclic set. Actually, one can give several characterisations of groupoids based on this set. It also gives a nice interpretation of well known notions of on groupoids: symplectic groupoids, duals of VB groupoids and in particular the Weinstein groupoid... Joint work with Claire Debord.

https://sites.google.com/view/julgfest

Orateur: Georges Skandalis (Université Paris Diderot)

11:15 → 12:10 Reciprocal hyperbolic elements in $\operatorname{PSL}_2(\mathbb{Z})$

An element $A$ in $\operatorname{PSL}_2(\mathbb{Z})$ is hyperbolic if $|\operatorname{Tr}(A)|>2$. The maximal virtually abelian subgroup of $\operatorname{PSL}_2(\mathbb{Z})$ containing $A$ is either infinite cyclic or infinite dihedral; say that $A$ is reciprocal if the second case happens ($A$ is then conjugate to its inverse). We give a characterization of reciprocal hyperbolic elements in $\operatorname{PSL}_2(\mathbb{Z})$ in terms of the continued fractions of their fixed points in $\operatorname{P}^1(\mathbb{R})$ (those are quadratic surds). Doing so we revisit results of P. Sarnak (2007) and C.-L. Simon (2022), themselves rooted in classical work by Gauss and Fricke \& Klein.

https://sites.google.com/view/julgfest

Orateur: Alain Valette (Université de Neuchâtel)

13:45 → 14:40 Tempiric representations, Connes-Kasparov, and pseudodifferential operators on symmetric spaces

Let $G$ be a real reductive group, connected for simplicity, with maximal compact subgroup $K$. The Connes-Kasparov isomorphism attaches a single parameter to most, but not all, of the components of the tempered dual of a real reductive group. That parameter is a shifted version of a highest weight for $K$, and every such parameter is attached to a unique component in the tempered dual. On the other hand, Vogan’s theory of minimal $K$-types associates to every component of the tempered dual of $G$ a finite collection of highest weights for $K$. These collections are disjoint, and every highest weight appears in one of them. Most of Vogan’s collections are singletons, but not all of them. The Connes-Kasparov and Vogan correspondences seem to be more similar than they are different, and in the online precursor to the current thematic program at IHP, Vogan asked whether they can be reconciled by adjusting the definition of the reduced C*-algebra of $G$? I shall discuss one answer, involving pseudodifferential operators, and some of the new questions that arise from that answer. This is joint work with Peter Debello.

https://sites.google.com/view/julgfest

Orateur: Nigel Higson (Penn State University)

14:45 → 15:40 On pseudodifferential calculi for subelliptic operators

Many constructions in representation theory and index theory are based upon elliptic differential operators. The work of Julg and Kasparov shows that subelliptic differential operators, such as the Berstein-Gelfand-Gelfand complex, are often required to make certain key constructions. In this talk, we will describe the tangent groupoid associated a very general class of such operators, namely operators of Helffer-Nourrigat type. Generalising ideas of Connes and Debord-Skandalis then allows us to prove the analytic properties of such operators.
(Joint work with Iakovos Androulidakis, Omar Mohsen and Erik Van Erp.)

https://sites.google.com/view/julgfest

Orateur: Robert Yuncken (Université de Lorraine)

16:00 → 16:55 Index theory on manifolds with a tangent Lie structure

In recent years there was a significant progress in the theory of pseudo-differential operators on filtered manifolds. I will introduce in my talk a wider class of manifolds which I call manifolds with a tangent Lie structure. I will explain a coarse approach to pseudo-differential theory which gives a simplified pseudo-differential calculus containing only operators of order 0 and negative order. This calculus easily leads to the Atiyah-Singer type index theorem for operators of order 0 on manifolds with a tangent Lie structure. For filtered manifolds this calculus agrees with the known H\"ormander and van Erp - Yuncken calculi, which allows to extend the index theorem to operators of any order.

https://sites.google.com/view/julgfest

Orateur: Gennadi Kasparov (Vanderbilt University)

17:00 → 17:55 From class field theory to zeta spectral triples

J'exposerai les résultats récents (< 1 an) de ma collaboration avec C. Consani
et H. Moscovici sur la compréhension conceptuelle de l'espace des classes d'adèles
comme extension de la théorie du corps de classe et la construction de triplets spectraux
donnant les zéros de zêta

https://sites.google.com/view/julgfest

Orateur: Alain Connes (IHES)

jeu. 6 février

10:00 → 11:00 The Baum-Connes conjecture for real reductive groups 7/10

Kasparov’s gamma element for SL(2,R)

Practical person’s guide to the Baum-Connes conjecture from the point of view of Kasparov theory: Kasparov’s representation ring (as an abelian group), the gamma element, Kasparov’s theorem for connected Lie groups, reduction to $\gamma=1$. First examples where $\gamma$ is equal to 1. Examples when $\gamma$ is not equal to 1. Julg’s proposal regarding uniformly bounded representations. Kasparov’s homotopy for SL(2,R).

Orateur: Nigel Higson (Penn State University)

11:00 → 12:00 The Baum-Connes conjecture for real reductive groups 8/10

Lafforgue’s Banach version of KK-theory

The C-algebra of a group, versus other convolution Banach algebras. Homotopies of unitary representations versus homotopies of Banach representations. Examples and action of the reduced C-algebra on $L^p$ spaces. Review of the role of compact operators in Kasparov theory. Dual pairs of Banach spaces and operators of compact type. Lafforgue’s version of Kasparov’s representation ring.

Orateur: Nigel Higson (Penn State University)

ven. 7 février

10:00 → 11:00 The Baum-Connes conjecture for real reductive groups 9/10

Banach modifications of the gamma element

The problem with Banach convolution algebras from the point of view of representation theory. The problem with tensor products and unconditional norms. Unconditional norms. Karoubi density. Reduction of the Baum-Connes conjecture to $\gamma=1$. A further technical reduction. The rapid decay property for the Harish-Chandra Schwartz algebra.

Orateur: Nigel Higson (Penn State University)

11:00 → 12:00 The Baum-Connes conjecture for real reductive groups 10/10

Rapid decay property and Lafforgue’s proof of the Baum-Connes conjecture

The gamma element from Witten’s perturbation of the de Rham complex. Transfer to weighted $L^2$ spaces. A homotopy from the gamma element to 1, almost. Putting everything together.

Orateur: Nigel Higson (Penn State University)

mar. 11 février

11:00 → 11:45 The noncommutative geometry of the space of positive elements in a C*-algebra

This talk delves into the fascinating differential geometry of positive elements of a $C^*$-algebra, highlighting its complexity, principal results, and depth. It also proposes an intriguing connection between this geometry and Connes' non-commutative approach, opening up new avenues for exploration in this field.

Orateur: Roberto Steven Vargas Peña (Estudiante de Doctorado)

14:00 → 14:45 Quantum groups meet graphs

Quantum groups and graph theory may seem like distant fields, yet intriguing connections arise when they intersect. After a gentle introduction to the topic, we explore these interactions through the study of quantum automorphism groups of block graphs. We conclude with some open problems and future directions. These results are parts of a common project with Amaury Freslon and Paul Meunier.

Orateur: Pegah Pournajafi (Collège de France)

15:00 → 15:45 First-order differential calculi and Laplacians on $q$-deformations of compact semisimple Lie groups

In this talk, we suggest a simple definition of Laplacian on a compact quantum group (CQG) associated with a first-order differential calculus (FODC) on it. Applied to the classical differential calculus on a compact Lie group, this definition yields classical Laplacians, as it should. Moreover, on the CQG $K_q$ arising from the $q$-deformation of a compact semisimple Lie group $K$, we can find many interesting linear operators that satisfy this definition, which converge to a classical Laplacian on $K$ as $q$ tends to 1. In light of this, we call them $q$-Laplacians on $K_q$ and investigate some of their operator theoretic properties. This work is based on the preprint arXiv:2410.00720.

Orateur: Dr Heon Lee (Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China)

16:00 → 17:00 Symmetry Breaking under Translations

Joint IHP - AIM RTNCG lecture

Abstract: I will discuss symmetry breaking for pairs of real forms of (GL(n, C), GL(n − 1, C)). We introduce the concept of "fences for the interlacing pattern," which refines the usual notion of "walls for Weyl chambers." We then present a theorem stating that multiplicity remains constant unless we cross these "fences." This approach is illustrated with examples of both tempered and non-tempered representations, along with a non-vanishing theorem of period integrals for pairs of reductive symmetric spaces.

Orateur: Prof. Toshiyuki Kobayashi (The University of Tokyo)

mer. 12 février

09:30 → 10:30 Unitary representations and spectral synthesis on contraction groups

The solution to Hilbert’s fifth problem dictates that every connected locally compact group is an inverse limit of connected Lie groups, and thus Lie theoretic techniques can be used in the study of connected locally compact groups. In contrast, totally disconnected locally compact (tdlc) groups cannot be approximated by Lie groups or algebraic group over tdlc fields, and their structure is not well understood, generally speaking. Modern research in topological group theory is largely focussed on understanding the class of tdlc groups. Furthermore, our understanding of the (unitary) representation theory and harmonic analysis of tdlc groups lags far behind that of connected locally compact groups.
In this talk, I will discuss recent progress on the harmonic analysis of (tdlc) contraction groups. Contraction groups can be viewed, in some sense, as analogues / generalisations of unipotent groups in the theory of tdlc groups. The focus will be on discussing recent progress on the unitary representation theory of these groups, and time pending, I will say a bit about spectral synthesis of certain convolution algebras on these groups.

Orateur: Max Carter (Université catholique de Louvain)

11:00 → 12:00 Lie groupoids and differential operators

In numerous PDE problems, differential operators naturally appear with some geometrical constraints. For instance, if $(M, \mathcal{F})$ is a foliated manifold one may be interested in operators acting longitudinally on the leaves, if $M$ has borders (or corners) one may ask operators to be well-behaved around the borders, on a manifold endowed with an action of a Lie group $G$ one may consider $G$-equivariant operators...
The aim of this talk is to explain how Lie groupoids provide a unified framework to study the analytic behaviour of all of these operators. This point of view allows to adapt the notions of principal symbols and ellipticity to these contexts and to define a well behaved pseudodifferential calculus in order to build parametrix. One can then use these operators to build classes in the K-theory of the $C^*$ algebras of the groupoids, as a first step towards Atiyah-Singer type theorems.

Orateur: Paul Le Breton (Paris cité)

14:00 → 15:00 Deformation to the Normal Cone, zoom action and distributions

The $C^*$-analysis of Lie groupoids provides powerful tools for understanding analysis on smooth manifolds and Lie groups. Deformations to the Normal Cone appear in the construction of fundamental instances of such Lie groupoids. In this talk we will discuss deformations to the normal cone, the zoom action and homogeneous distributions on a DNC. If time permits, we will also discuss applications to analysis on manifolds such as the pseudodifferential calculus \emph{à la} Van Erp and Yuncken.

Orateur: Moudrik Chamoux

15:30 → 16:30 Automorphisms of pseudodifferential operators on Lie groupoids

The study of pseudodifferential operators on Lie groupoids offers a natural extension of classical analysis on smooth manifolds, encompassing singular spaces. Moreover, pseudodifferential operators can be generalized in the classical setting by Fourier integral operators. A key result in this context is Egorov’s theorem: conjugation by a Fourier integral operator preserves pseudodifferential operators. Duistermaat and Singer later established a converse: any automorphism of the filtered algebra of classical pseudodifferential operators can be realized as conjugation by a Fourier integral operator.
In this talk, I will discuss a generalization of this result in the setting of Lie groupoids. This involves a detailed examination of the symplectic structure of the cotangent groupoid, which is fundamental for constructing Fourier integral operators in this context.

Orateur: Lucas Lemoine (Université Paris-Est Créteil)

jeu. 13 février

19:00 → 20:30 Conférence carte blanche « Quand la symétrie prédit »

This talk will be delivered in French at Maison Henri Poincaré. Entrance is free of charge, but registration is mandatory, at: https://billetterie-maison-poincare.ihp.fr/evenements/conferences-cartes-blanche/quand-la-symetrie-predit

Dans la vie courante comme en sciences, l'idée de symétrie évoque l’harmonie, l'équilibre. Depuis deux siècles, en mathématiques, les symétries se quantifient : des outils mathématiques précis, les groupes, permettent de mieux comprendre les symétries qui nous sont familières et d’en découvrir de nouveaux types. Depuis un siècle, on sait utiliser ces outils pour aller au-delà de la contemplation de l’harmonie : les groupes, et ce qu’on appelle leurs représentations, ont permis de faire des prédictions concrètes et inattendues dans des domaines variés — de la physique quantique à la théorie des nombres. Cette conférence évoque quelques épisodes de l’histoire des représentations de groupes, et esquisse quelques espoirs que les scientifiques d’aujourd’hui placent en elles.

Orateur: Alexandre Afgoustidis (CNRS & Institut Élie Cartan de Lorraine)

lun. 17 février

14:00 → 15:00 Structure of tempered homogeneous spaces 1/3 - Dynamical approach

The three lectures introduce recent theories of tempered spaces, and I plan to provide an overview of these topics, using plenty of elementary examples to make the basic concepts and key ideas more accessible.
In the first lecture, I will review basic concepts such as tempered unitary representations of real reductive groups, like GL(n, R), as well as “tempered spaces” and “tempered subgroups”. I will begin with some geometric observations of group actions, including the properness criterion for reductive homogeneous spaces. Subsequently, I will introduce a “quantification” of proper actions and incorporate a dynamical approach into analytic representation theory, including the temperedness criterion for homogeneous spaces, which was developed recently by Y. Benoist and the speaker, drawing on the Cowling-Haagerup-Howe theory and other related ideas.

Orateur: Prof. Toshiyuki Kobayashi (The University of Tokyo)

mar. 18 février

14:30 → 15:30 The Dolbeault-Dirac operator on the irreducible quantum flag manifolds

In this talk I will comment about some aspects in the construction of the Dolbeault-Dirac operator d+d* associated to some type of quantum homogeneous spaces generalizing the classical construction of the Rham complex of smooth manifolds. I will introduce a quantum version of the Bernstein-Gelfand-Gelfand resolution of irreducible quantum flag manifolds in order to dualize it in some way to get the Dolbeault complex and define the Dolbeault-Dirac operator. If time permits, I will give the example of the irreducible quantum flag of type B_2 for which the Dolbeault-Dirac operator leads to a spectral triple in the sense of Connes, also I will give some comments on the case of the quantum Grassmanian Gr(2,4) which is an ongoing project joint with E. Wagner.

Orateur: Fredy Diaz Garcia (National Autonomous University of Mexico)

16:00 → 17:00 Parthasarathy formulas and beyond

Joint IHP - AIM RTNCG lecture

Abstract: Dirac operators have played an important role in representation theory, originating in the work Parthasarathy, who proved a formula for the square of a Dirac operator on a Riemannian symmetric space, and used it to realize discrete series representations of semi simple Lie groups as L2 spaces of harmonic spinors.

Later on, Huang and Pandzic, observing that the Parthasarathy formula implies that the square of a Dirac operator is central, developed the theory of Dirac cohomology, and derived property of admissible representations that may not be discrete series. In this talk we will first illustrates these results using Parthasarathy formula, to explain how Vogan classification of tempered representations by their lowest K types, fits into the computation the G index of the Dirac operator.

We will next introduce an analogue of the Dirac operator for symplectic homogeneous spaces, and state a Parthasarathy formula in this context, when an invariant polarization exists and a symmetry condition holds. Generalizations of the Parthasarathy formula were proved by Kostant, introducing a cubic term, when the quadratic homogeneous space is not symmetric. We will show that no cubic terms may lead to any corresponding formula in the case of the symplectic Dirac operator.

Orateur: Nicolas Prudhon (université de Lorraine)

jeu. 20 février

11:00 → 12:00 Twisted Ruelle zeta functions, complex-valued analytic torsion and the Fried's conjecture - 1/3

In this mini course, we will introduce the dynamical zeta functions of Ruelle and Selberg for hyperbolic manifolds or more generally for locally symmetric spaces of rank one.
We will present the so called Fried 's conjecture, i.e., a conjectural equality between the Ruelle zeta function at zero and a spectral invariant, the analytic torsion. In the second part of the mini course, we will talk about this problem, in the setting of a non-unitary representation of the lattice we consider. We will see how one can use the trace formula to study this problem and also we will present the definition of the refined analytic torsion and the Cappell-Miller torsion. Time depending, we will discuss some further problems related with these objects.

Orateur: Polyxeni Spilioti

14:00 → 15:00 Structure of tempered homogeneous spaces 2/3 - Combinatorics approach

The criterion for tempered spaces, explained in the first lecture, is computable. In this lecture, I will explain how this criterion leads to the classification theory of non-tempered reductive homogeneous spaces by breaking it down into several steps. The technical methods used in the second lecture differ from the dynamical approach presented in the first lecture. Our approach relies on elementary results from finite-dimensional representations and some combinatorics of convex polyhedral cones.

Orateur: Prof. Toshiyuki Kobayashi (The University of Tokyo)

ven. 21 février

09:30 → 10:30 Twisted Ruelle zeta functions, complex-valued analytic torsion and the Fried's conjecture - 2/3

In this mini course, we will introduce the dynamical zeta functions of Ruelle and Selberg for hyperbolic manifolds or more generally for locally symmetric spaces of rank one.
We will present the so called Fried 's conjecture, i.e., a conjectural equality between the Ruelle zeta function at zero and a spectral invariant, the analytic torsion. In the second part of the mini course, we will talk about this problem, in the setting of a non-unitary representation of the lattice we consider. We will see how one can use the trace formula to study this problem and also we will present the definition of the refined analytic torsion and the Cappell-Miller torsion. Time depending, we will discuss some further problems related with these objects.

Orateur: Polyxeni Spilioti

11:00 → 12:00 Twisted Ruelle zeta functions, complex-valued analytic torsion and the Fried's conjecture - 3/3

In this mini course, we will introduce the dynamical zeta functions of Ruelle and Selberg for hyperbolic manifolds or more generally for locally symmetric spaces of rank one.
We will present the so called Fried 's conjecture, i.e., a conjectural equality between the Ruelle zeta function at zero and a spectral invariant, the analytic torsion. In the second part of the mini course, we will talk about this problem, in the setting of a non-unitary representation of the lattice we consider. We will see how one can use the trace formula to study this problem and also we will present the definition of the refined analytic torsion and the Cappell-Miller torsion. Time depending, we will discuss some further problems related with these objects.

Orateur: Polyxeni Spilioti

14:00 → 15:00 Structure of tempered homogeneous spaces 3/3 - Limit algebras

Recently, surprising and intriguing connections have been observed between the concept of "tempered spaces" for unitary representations and various other areas of mathematics. In this lecture, we will explore different aspects of tempered spaces from the perspectives of topology and geometry, including limit algebras (collapsing Lie algebras) and geometric quantization.

Orateur: Prof. Toshiyuki Kobayashi (The University of Tokyo)

lun. 24 février

09:00 → 12:00 Workshop: Tempered representations and K-theory

See: https://indico.math.cnrs.fr/event/10857

15:00 → 16:00 Workshop: Tempered representations and K-theory

See: https://indico.math.cnrs.fr/event/10857

mar. 25 février

09:30 → 12:00 Workshop: Tempered representations and K-theory

See: https://indico.math.cnrs.fr/event/10857/

14:30 → 17:00 Workshop: Tempered representations and K-theory

See: https://indico.math.cnrs.fr/event/10857/

mer. 26 février

09:30 → 12:00 Workshop: Tempered representations and K-theory

See: https://indico.math.cnrs.fr/event/10857/

jeu. 27 février

09:30 → 12:00 Workshop: Tempered representations and K-theory

See: https://indico.math.cnrs.fr/event/10857/

15:00 → 16:00 Workshop: Tempered representations and K-theory

See: https://indico.math.cnrs.fr/event/10857/

16:30 → 17:30 Topos and noncommutative geometry: Two perspectives on spaces and numbers

Noncommutative geometry and the notion of topos are two mathematical concepts
that provide complementary perspectives on the structure of a space. In this talk, I will
begin by explaining, as simply as possible, these two concepts and what makes them
unique. The originality of noncommutative geometry can be directly perceived through
the existence of an intrinsic time evolution of a noncommutative space. The originality
of toposes can similarly be perceived through the intuitionistic logic associated with a
topos. It is the metric structure, embodied by a representation—as operators in Hilbert
space—of coordinates and the length element, that allows noncommutative geometry
to engage with reality, namely the structure of space-time at the infinitesimally small
scale as revealed by contemporary physics through the Standard Model. As for
toposes, it is the additional structure of a sheaf of algebras that enables geometry to
manifest beyond topology.
In the second part of the talk, I will explain how the spectrum of the ring of integers
can be understood through these two geometric lenses. The connection between
these two approaches rests on an extension of class field theory that sheds light on
the analogy established by Mumford and Mazur between knots and prime numbers.
The spectral perception of the ring of integers naturally emerges from the study of
the zeros of the Riemann zeta function, thereby revealing deep structures at the
interface of arithmetic, topology, and geometry

Orateur: Alain Connes (IHES)

ven. 28 février

09:30 → 12:00 Workshop: Tempered representations and K-theory

See: https://indico.math.cnrs.fr/event/10857/

lun. 3 mars

09:00 → 10:30 Mini-course: The Bernstein center - 1/3

In this series of lectures, we will explain J. Bernstein's description of the center of the category of smooth representations of reductive $p$-adic groups as rational functions on the variety of irreducible representations. The course is intended for PhD students and non-specialists.

Orateur: David Renard (Ecole Polytechnique)

11:00 → 12:00 $\ell$-Modular blocks of $\mathrm{SL}_n$

Reduction to depth zero is a promising approach for understanding $\ell$-modular blocks of $p$-adic groups when $\ell$ differs from $p$. In this talk, I will introduce $\ell$-modular blocks of $\mathrm{SL}_n$ from this perspective. We will explore the technical challenges in associating an $\ell$-modular block with a depth-zero block and consider a natural candidate for this potential connection. Toward the end, I will also discuss related topics on both the automorphic and Galois sides.

Orateur: Peiyi Cui (Morningside Center of Mathematics)

14:00 → 15:00 Representations of $p$-adic groups in arbitrary residue characteristic

This talk is an introduction to the representation theory of p-adic groups aimed at all participants of the trimester program. I will survey the current state of the art and include new developments of the last six months. We will focus on two crucial aspects:
First, I will provide an overview of the construction of all so called supercuspidal representations, which are the building blocks for all representations. For more than 20 years it remained open to extend Yu's general construction to the case p=2, and I will sketch what makes this case so special and how we could overcome the obstacles in my recent joint work with David Schwein.
Second, we will study the structure of the whole category of representations of p-adic groups in terms of these supercuspidal
representations, and I will explain how two recent preprints with Jeffrey Adler, Manish Mishra and Kazuma Ohara allow us to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or are at least easier to achieve.

Orateur: Jessica Fintzen (University of Bonn)

15:00 → 16:00 Bounding Harish-Chandra characters

Let $G$ be a connected reductive algebraic group over a $p$-adic local field $F$. We study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular semisimple element of $G(F)$ as $\pi$ varies among supercuspidal representations of $G(F)$. Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{\deg(\pi)}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $G(F)$ whose central character is unitary and the formal degree of $\pi$ tends to infinity. I will sketch the proof that for $G$ semisimple the trace character is uniformly bounded on $\gamma$ under the assumption, which is believed to hold in general, that all irreducible supercuspidal representations of $G(F)$ are compactly induced from an open CVcompact modulo center subgroup. If time allows I could also discuss progress on optimizing the bound.

Orateur: Anna Szumowicz (IMPAN)

16:30 → 17:30 $p$-modular Iwahori-Hecke algebras and their simple modules for the $p$-adic metaplectic group $\widetilde{\mathrm{SL}}_2(F)$

Orateur: Dr Ramla ABDELLATIF (LAMFA - Université de Picardie Jules Verne)

mar. 4 mars

09:30 → 10:30 Asymptotics, mostly for $SL_2$

Let F be a non Archimedean local field and let p be its residue characteristic. Let R be an algebraically closed field with char(R) different from p. We consider a connected reductive group G over F, and look at irreducible smooth R-representations of the locally pro-p group G=G(F). Classifying all such representations up to isomorphism is very involved, but their behaviour under restriction to small enough open pro-p subgroups is expected to be more uniform. When R=C and char(F)=0, that can be obtained from Harish-Chandra's germ expansion. One can hope for a similar control for general R and char(F)>=0. This has consequences for the asymptotics of fixed points under congruence subgroups of G. We shall survey what is known for G=GL(N) and G=SL(2) (Joint work with Vignéras) and what it suggests for the general case.

Orateur: Guy Henniart (Université Paris-Saclay)

10:30 → 10:45 Group photo

11:00 → 12:30 Mini-course: The Bernstein center - 2/3

In this series of lectures, we will explain J. Bernstein's description of the center of the category of smooth representations of reductive $p$-adic groups as rational functions on the variety of irreducible representations. The course is intended for PhD students and non-specialists.

Orateur: David Renard (Ecole Polytechnique)

14:00 → 15:30 Mini-course: The Bernstein center - 3/3

In this series of lectures, we will explain J. Bernstein's description of the center of the category of smooth representations of reductive $p$-adic groups as rational functions on the variety of irreducible representations. The course is intended for PhD students and non-specialists.

Orateur: David Renard (Ecole Polytechnique)

16:00 → 17:00 On the unitary dual of a $p$-adic group and the Aubert involution

A significant aspect of representation theory research is determination of the unitary dual of a reductive algebraic group over a local non-archimedean field. We will look at the unitary dual of $p$-adic group $SO(7)$ with support on minimal parabolic subgroup. There is the conjecture stating that the Aubert involution preserves unitarity confirmed the conjecture for that case. This work is supported (in part) by the Croatian Science Foundation under the project number HRZZ-IP-2022-10-4615.

Orateur: Darija Brajković Zorić (School of Applied Mathematics and Informatics, Josip Juraj Strossmayer University of Osijek)

mer. 5 mars

09:30 → 10:30 A multiplicity one theorem for general Spin groups

There are many ingredients needed to prove a local Gan-Gross-Prasad conjecture. In this talk, we will focus on the first ingredient: a multiplicity at most one theorem. Aizenbud, Gourevitch, Rallis and Schiffmann proved a multiplicity at most one theorem for restrictions of irreducible representations of certain p-adic classical groups and Waldspurger proved the same theorem for the special orthogonal groups. We will discuss work that establishes a multiplicity at most one theorem for restrictions of irreducible representations for a non-classical group, the general spin group. This is joint work with Shuichiro Takeda.

Orateur: Melissa Emory (Oklahoma State University)

11:00 → 12:00 Non-basic rigid packets for discrete L-parameters

In this talk I'll propose a new way to organize discrete compound L-packets of a p-adic group G. Our packets include representations not only of inner twists of G but also of its elliptic twisted Levi subgroups, conjecturally realizing some version of loop Deligne–Lusztig induction or Yu's construction. The main idea is to replace Kottwitz's set with the non-basic cohomology of the Kaletha gerbe. This new cohomology set is much richer but turns out to carry many of the same structures as B(G), such as the Kottwitz and Newton maps. On the Galois side, we define a rigid enhancement for L-parameters and show that the resulting refined local Langlands conjectures reduce, à la Bertoloni Meli–Oi, to the basic rigid conjectures of Kaletha. This work is joint with Peter Dillery.

Orateur: David Schwein (University of Bonn)

jeu. 6 mars

09:00 → 10:30 Mini-course: Hochschild homology of reductive $p$-adic groups - 1/3

The goal of this mini-course is to explain a description of the Hochschild homology of the Hecke algebra of a reductive $p$-adic group, in spectral terms related to the Bernstein center. The lectures will start quite general, and will gradually specialize towards $p$-adic groups. No prior knowledge of Hochschild homology is required.
Lecture 1: Introduction to Hochschild homology and twisted crossed product algebras
Lecture 2: Representation theory and Hochschild homology of graded Hecke algebras
Lecture 3: Hochschild homology of algebras associated to $p$-adic groups

Orateur: Maarten Solleveld (Radboud Universiteit Nijmegen)

11:00 → 12:00 Mini-course: Hochschild homology of reductive $p$-adic groups - 2/3

The goal of this mini-course is to explain a description of the Hochschild homology of the Hecke algebra of a reductive $p$-adic group, in spectral terms related to the Bernstein center. The lectures will start quite general, and will gradually specialize towards $p$-adic groups. No prior knowledge of Hochschild homology is required.
Lecture 1: Introduction to Hochschild homology and twisted crossed product algebras
Lecture 2: Representation theory and Hochschild homology of graded Hecke algebras
Lecture 3: Hochschild homology of algebras associated to $p$-adic groups

Orateur: Maarten Solleveld (Radboud Universiteit Nijmegen)

14:00 → 15:00 Cuspidal $l$-modular representations of $p$-adic $GL(n)$ distinguished by a Galois involution

Let $E/F$ be a quadratic extension of $p$-adic fields. A smooth representation of ${\rm GL}_n(E)$ is said to be distinguished by ${\rm GL}_n(F)$ if it carries a non-zero ${\rm GL}_n(F)$-invariant linear form. Distinguished complex representations have been extensively studied: there is in particular a full classification of distinguished generic complex representations. The case of $l$-modular representations (that is, with coefficients in a field whose characteristic is a prime number $l$ different from $p$) is much less well understood. In this talk, I will discuss the case of cuspidal $l$-modular representations (for $p$ odd). This is a joint work with Robert Kurinczuk and Nadir Matringe.

Orateur: vincent secherre (Université de Versailles St-Quentin)

15:00 → 16:00 Endoscopic character identities for depth-zero supercuspidal representations of SL_2

We consider the depth-zero supercuspidal $L$-packets of $\mathrm{SL}_2$. With the aid of the classical character formulas of Sally-Shalika, we prove the endoscopic character identities. Our account fills a gap in the extensive literature on $\mathrm{SL}_2$. Joint work with Anne-Marie Aubert, see arXiv:2410.20183v2

Orateur: Roger Plymen

ven. 7 mars

09:00 → 10:30 Mini-course: Hochschild homology of reductive $p$-adic groups - 3/3

The goal of this mini-course is to explain a description of the Hochschild homology of the Hecke algebra of a reductive $p$-adic group, in spectral terms related to the Bernstein center. The lectures will start quite general, and will gradually specialize towards $p$-adic groups. No prior knowledge of Hochschild homology is required.
Lecture 1: Introduction to Hochschild homology and twisted crossed product algebras
Lecture 2: Representation theory and Hochschild homology of graded Hecke algebras
Lecture 3: Hochschild homology of algebras associated to $p$-adic groups

Orateur: Maarten Solleveld (Radboud Universiteit Nijmegen)

11:00 → 12:00 Rankin-Selberg integrals and double flag varieties

The Rankin–Selberg method is a classical approach for constructing integral representations of L-functions, facilitating their meromorphic continuation and the establishment of functional equations. In this talk, we present a novel perspective that interprets Rankin–Selberg integrals within the framework of double flag varieties of finite type. We introduce the concept of strongly tempered spherical varieties of Rankin–Selberg type and, leveraging the classification by He, Nishiyama, Ochiai, and Oshima, systematically identify all Rankin–Selberg integrals associated with these varieties. This framework enables a unified treatment of local zeta integrals and the gamma factors for this family. As an application, we can derive complete L-functions corresponding to these Rankin–Selberg integrals and establish local multiplicity formulas via the local relative trace formula for certain strongly tempered spherical varieties of Gan-Gross-Prasad type over p-adic fields.

Orateur: Lei Zhang (National University of Singapore)

14:00 → 15:00 Geometric analogues to local Arthur packets: $p$-adic classical groups

Arthur packets are sets containing irreps that form constituents of important classes of automorphic representations. ABV-packets, named after Jeff Adams, Dan Barbasch, and David Vogan, are proposed generalizations to local Arthur packets for archimedean groups. They arise out of a geometric perspective on the local Langlands correspondence involving microlocal analysis on moduli spaces of Langlands parameters. Vogan gave an analogous description for the non-archimedean case, which was then adapted by Cunningham et al. The proposal that an ABV-packet for a Langlands parameter of Arthur-type coincides with the corresponding local Arthur packet is stated as Vogan's conjecture by Cunningham et al in their adaptation. This talk aims to report on the progress of Vogan's conjecture for $p$-adic general linear, odd special orthogonal, and symplectic groups.

Orateur: Mishty Ray (University of Calgary)

lun. 10 mars

09:30 → 10:30 Progress on the local Gan-Gross-Prasad conjecture

The classical branching problem studies the restriction of an irreducible representation to a compact subgroup. The works of Gross-Prasad and Gan-Gross-Prasad generalized this framework into a conjecture for classical groups over local fields of characteristic zero. The first breakthrough was achieved by Waldspurger in the non-Archimedean special orthogonal cases. Since then, various approaches have been developed, leading to the complete proof of the conjecture in all cases.

In this presentation, I will introduce an approach that applies to both unitary and non-unitary cases, Archimedean and non-Archimedean, as well as Bessel and Fourier-Jacobi cases. This approach, based on the foundational works of Waldspurger, Mœglin-Waldspurger, and Gan-Ichino, utilizes the trace formula, endoscopy, the multiplicity formula, and the Theta correspondence. The development of this approach includes some of my work, as well as collaborative efforts with Luo, work in collaboration with Chen and Zou, and work in collaboration with Jiang, Liu, and Zhang.

Orateur: Cheng Chen (University of Minnesota, Twin Cities)

10:45 → 12:15 Mini-course: The theta correspondence for classical groups over finite fields of odd characteristic 1/3

The theta correspondence for classical groups over finite fields of odd characteristic is defined via the Weil representation of a finite symplectic group.
It maps a given irreducible representation of a group $G$ to a finite collection of irreducible representations of a group $G'$, such that $(G,G')$ is a dual pair.

I will first recall the definition of the correspondence and some its useful properties. Next, I will provide a full and explicit description of the theta correspondence via a conjecture
that I have formulated, and proved in the case of linear and unitary dual pairs, with J. Michel and R. Rouquier, and which has been recently established in all cases by Pan and by Ma, Qiu and Zou, independently.

I will later explain several ways to extract a bijective correspondence, including the construction by Gurevich and Howe of the eta correspondence, and give some applications. I will also describe the links between the theta correspondence and the finite analogue of the Gan-Gross-Prasad problem, which can be viewed as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh.

Orateur: Prof. Anne-Marie Aubert (IMJ-PRG CNRS)

14:00 → 15:30 Mini-course: Minimal representations and applications 1/3

I will define minimal representations of $p$-adic groups and explain how one can work with them. As an application, using minimal representations, I will decompose certain Weil representations attached to multiplicity-free symplectic representations as defined and classified by Knop. These results are consistent with predictions in the relative Langlands program of Ben-Zvi, Sakellaridis and Venkatesh.

Orateur: Gordan Savin (University of Utah)

15:45 → 16:45 Theta Correspondence between principle series and its geometrization

Consider the reductive dual pairs consisting of a symplectic group and an orthogonal group. The Harish-Chandra series of these two groups correspond to each other under the theta correspondence. In collaboration with Congling Qiu and Jialiang Zou, we explicitly computed this correspondence by analyzing the relevant Hecke algebra bimodules and applying a Tits deformation argument. This approach provides an alternative proof of Aubert-Michel-Rouquier's conjecture, which was initially settled by Shu-Yen Pan. In this presentation, we explore the geometrization of this construction. Consequently, we have derived a new description of the theta correspondence in terms of the Springer theory. This is a joint work with Congling Qiu, Jialiang Zou, and Zhiwei Yun.

Orateur: JIAJUN MA (Xiamen University)

17:00 → 18:00 Modular local theta correspondence

The local theta correspondence over a non-Archimedean local field of residual characteristic p asserts a bijection between (subsets of) irreducible complex representations of two reductive groups forming a dual pair in a symplectic group.
In this talk, I will explain how this theory can be generalized to $l$-modular representations — i.e. when the coefficient field has positive characteristic $l$, distinct from $p$. Provided that l is sufficiently large relative to the size of the dual pair, this generalisation also results in a bijection, which we refer to as the $l$-modular theta correspondence. However, for certain values of~$l$, such a bijection fails to hold.

Orateur: Justin Trias (University of East Anglia)

mar. 11 mars

09:30 → 10:30 Theta correspondence as an induction functor for group C*-algebras

In recent works, we have exhibited a C*-algebraic approach to theta correspondence. I will try to explain this approach, its connection to Jian-Shu Li's explicit theta lifting method and will try to convince representation theorists that such an approach is interesting. Joint works with B. Mesland (Leiden) and M. Goffeng (Lund).

Orateur: Haluk Sengun (University of Sheffield)

10:45 → 12:15 Mini-course: The theta correspondence for classical groups over finite fields of odd characteristic 2/3

The theta correspondence for classical groups over finite fields of odd characteristic is defined via the Weil representation of a finite symplectic group.
It maps a given irreducible representation of a group $G$ to a finite collection of irreducible representations of a group $G'$, such that $(G,G')$ is a dual pair.

I will first recall the definition of the correspondence and some its useful properties. Next, I will provide a full and explicit description of the theta correspondence via a conjecture
that I have formulated, and proved in the case of linear and unitary dual pairs, with J. Michel and R. Rouquier, and which has been recently established in all cases by Pan and by Ma, Qiu and Zou, independently.

I will later explain several ways to extract a bijective correspondence, including the construction by Gurevich and Howe of the eta correspondence, and give some applications. I will also describe the links between the theta correspondence and the finite analogue of the Gan-Gross-Prasad problem, which can be viewed as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh.

Orateur: Prof. Anne-Marie Aubert (IMJ-PRG CNRS)

14:00 → 15:30 Mini-course: Minimal representations and applications 2/3

I will define minimal representations of $p$-adic groups and explain how one can work with them. As an application, using minimal representations, I will decompose certain Weil representations attached to multiplicity-free symplectic representations as defined and classified by Knop. These results are consistent with predictions in the relative Langlands program of Ben-Zvi, Sakellaridis and Venkatesh.

Orateur: Gordan Savin (University of Utah)

16:00 → 17:00 New supercuspidal representations from the Weil representation in characteristic two

Supercuspidal representations are the basic building blocks in the representation theory of reductive p-adic groups, and there is intense interest in constructing these representations explicitly. The construction problem faces many additional obstacles in residue characteristic two, however, and even for classical groups, our knowledge there is incomplete. In this talk, based on joint work with Jessica Fintzen, I'll explain how to overcome one such obstacle: the exceptional behavior of the Weil representation in characteristic two.

Orateur: David Schwein (University of Bonn)

mer. 12 mars

09:30 → 10:30 Automorphic kernel and Langlands conjecture

I will discuss constructions of automorphic kernel functions, with which the integral transforms will conjecturally yield the analytic properties of automorphic L-functions and the certain functoriality as expected by the Langlands conjectures. The known cases includes the theta correspondence and certain Rankin-Selberg global zeta integrals".

Orateur: Dihua Jiang (School of Mathematics, University of Minnesota)

10:45 → 12:15 Mini-course: The theta correspondence for classical groups over finite fields of odd characteristic 3/3

The theta correspondence for classical groups over finite fields of odd characteristic is defined via the Weil representation of a finite symplectic group.
It maps a given irreducible representation of a group $G$ to a finite collection of irreducible representations of a group $G'$, such that $(G,G')$ is a dual pair.

I will first recall the definition of the correspondence and some its useful properties. Next, I will provide a full and explicit description of the theta correspondence via a conjecture
that I have formulated, and proved in the case of linear and unitary dual pairs, with J. Michel and R. Rouquier, and which has been recently established in all cases by Pan and by Ma, Qiu and Zou, independently.

I will later explain several ways to extract a bijective correspondence, including the construction by Gurevich and Howe of the eta correspondence, and give some applications. I will also describe the links between the theta correspondence and the finite analogue of the Gan-Gross-Prasad problem, which can be viewed as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh.

Orateur: Prof. Anne-Marie Aubert (IMJ-PRG CNRS)

14:00 → 15:30 Mini-course: Local theta correspondence 1/3

This mini-course on local theta correspondence will be mainly concerned with the Archimedean theory, and will consists of three lectures. In the first lecture, I will explain its basic theory, including the Howe duality theorem, and the conservation relations. I will also discuss some examples. In the second lecture, I will explain how some of the basic invariants of representations behave under the local theta correspondence. The final lecture will be about applications to unitary representation theory.

Orateur: Chen-Bo Zhu (National University of Singapore)

15:45 → 16:45 The Weil representation for a finite field of characteristic two

The Weil representation is relatively well understood for local fields or finite fields of odd characteristics. In characteristic two, even in the case of a finite field, it is still not understood.
In this talk, we will present an explicit construction of the Weil representation for a finite field of characteristic two.
After defining the Heisenberg group in this setting, we construct the Weil representation of a two-fold covering of the pseudo-symplectic group.
We obtain explicit formulas for this representation, its character and the associated cocycle.
The pseudo-symplectic group is an extension of the orthogonal group, which is smaller than the symplectic group. Therefore, for the field $\mathbb{F}_2$, using the ring $\mathbb{Z}/4\mathbb{Z}$, we will provide a second construction of the Weil representation, defined on a covering of the affine symplectic group. All along, we will illustrate our results with the example of a two-dimensional vector space over $\mathbb{F}_2$.

Orateur: Aurélie Paull (Université de Lorraine)

jeu. 13 mars

09:30 → 10:30 The theta correspondence over $p$-adic fields in terms of Bernstein blocks of enhanced $L$-parameters

I will describe a partition, obtained with A. Moussaoui and M. Solleveld, of enhanced $L$-parameters for $p$-adic reductive groups in -- Galois analogues of -- Bernstein blocks.

Next, in the case of classical groups, I will show the compatibility of the theta correspondence with this partition, and how it allow to obtain an explicit description of the correspondence.

Orateur: Prof. Anne-Marie Aubert (IMJ-PRG CNRS)

AM25_W2--new.pdf

10:45 → 12:15 Mini-course: Local theta correspondence 2/3

This mini-course on local theta correspondence will be mainly concerned with the Archimedean theory, and will consists of three lectures. In the first lecture, I will explain its basic theory, including the Howe duality theorem, and the conservation relations. I will also discuss some examples. In the second lecture, I will explain how some of the basic invariants of representations behave under the local theta correspondence. The final lecture will be about applications to unitary representation theory.

Orateur: Chen-Bo Zhu (National University of Singapore)

14:00 → 15:30 Mini-course: Minimal representations and applications 3/3

I will define minimal representations of $p$-adic groups and explain how one can work with them. As an application, using minimal representations, I will decompose certain Weil representations attached to multiplicity-free symplectic representations as defined and classified by Knop. These results are consistent with predictions in the relative Langlands program of Ben-Zvi, Sakellaridis and Venkatesh.

Orateur: Gordan Savin (University of Utah)

15:45 → 16:45 Symmetry breaking operators and the Howe correspondence for dual pairs with one member compact

We consider Howe correspondence for dual pairs with one member compact. Given an irreducible representation $\Pi$ of the compact member, we study the projection operator $P_\Pi$ of the Weil representation onto its $\Pi$-isotopic component. These operators are examples of symmetry breaking operators in the sense of Kobayashi. Guided by Hermann Weyl’s integration formula, we study orbital integrals on the underlying symplectic space. They are needed to compute the Weyl symbols of $P_\Pi$. Our main example will be the dual pair of two compact unitary groups. In this case, we recover by purely analytic methods the existence of Howe correspondence for compact unitary groups, or equivalently, Weyl’s First Fundamental Theorem for Classical Invariant Theory for complex general linear groups. This is joint work with Mark McKee and Tomasz Przebinda (University of Oklahoma).

Orateur: Angela Pasquale (Institut Elie Cartan de Lorraine (IECL, UMR 7502), Metz)

17:00 → 18:00 Unipotent Arthur packets for real classical groups

Suppose that $G$ is a real symplectic group or a real even special orthogonal group. We explicitly construct all representations in an arbitrary unipotent Arthur packet of $G$, by using parabolic induction and iterated theta lift. The talk is based on a joint work with Dan Barbasch, Jia-Jun Ma and Chen-Bo Zhu, and a joint work with Taiwang Deng, Chang Huang, and Bin Xu.

Orateur: Binyong Sun (Zhejiang University)

ven. 14 mars

09:30 → 10:30 Howe duality for Lie superalgebras

As suggested in "Remarks on classical invariant theory”, the Howe duality in the symplectic group should be extendable to the orthosymplectic supergroup, with the Weil representation replaced by the spinor-oscillator representation of the supergroup SpO. In my talk, I will explain what is known about the duality in the supercase, present some of the results obtained recently with H. Salmasian, and give an explicit description of the duality for the dual pair $({\mathrm{SpO}}(2n|1), {\mathrm{osp}}(2|2))$.

Orateur: Allan Merino (Simmons University)

10:45 → 12:15 Mini-course: Local theta correspondence 3/3

This mini-course on local theta correspondence will be mainly concerned with the Archimedean theory, and will consists of three lectures. In the first lecture, I will explain its basic theory, including the Howe duality theorem, and the conservation relations. I will also discuss some examples. In the second lecture, I will explain how some of the basic invariants of representations behave under the local theta correspondence. The final lecture will be about applications to unitary representation theory.

Orateur: Chen-Bo Zhu (National University of Singapore)

lun. 17 mars

10:00 → 10:45 Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 1/4 (pt.1)

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$.

Lecture 1: Nets of operator algebras and AQFT.

We start with the translation from nets of operator algebras to nets of real subspaces, based on modular theory. We introduce real standard subspaces, discuss the Tomita-Takesaki Theorem as a key result from the modular theory of operator algebras and then describe axioms for nets of real subspaces ${\sf H}({\mathcal O})$ in a unitary representation of a Lie group. These are structures than can be explored completely from the perspective of the geometry of homogeneous spaces and unitary representations.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Orateur: Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

11:00 → 11:45 Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 1/4 (pt.2)

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$.

Lecture 1: Nets of operator algebras and AQFT.

We start with the translation from nets of operator algebras to nets of real subspaces, based on modular theory. We introduce real standard subspaces, discuss the Tomita-Takesaki Theorem as a key result from the modular theory of operator algebras and then describe axioms for nets of real subspaces ${\sf H}({\mathcal O})$ in a unitary representation of a Lie group. These are structures than can be explored completely from the perspective of the geometry of homogeneous spaces and unitary representations.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Orateur: Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

14:30 → 16:00 Problem session

This session will be devoted to exercises and complements related to the mini-course "Algebraic Quantum Field Theory and causal homogeneous spaces" by K.-H. Neeb.

Orateur: Tobias Simon (Friedrich-Alexander Universität Erlangen-Nürnberg (FAU))

mar. 18 mars

10:00 → 10:45 Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 2/4 (pt.1)

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$.

Lecture 2: Euler elements and causal homogeneous spaces.

We explore which specific structures we need on the homogeneous space $M = G/H$ and the Lie group $G$, so that a rich supply of nets may exist. In particular, we explain how Euler elements of Lie algebras (elements defining 3-gradings) enter the picture as candidates of generators of modular groups. This leads to several families of causal homogeneous spaces such as compactly and non-compactly causal symmetric spaces and causal flag manifolds.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Orateur: Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

11:00 → 11:45 Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 2/4 (pt.2)

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$.

Lecture 2: Euler elements and causal homogeneous spaces.

We explore which specific structures we need on the homogeneous space $M = G/H$ and the Lie group $G$, so that a rich supply of nets may exist. In particular, we explain how Euler elements of Lie algebras (elements defining 3-gradings) enter the picture as candidates of generators of modular groups. This leads to several families of causal homogeneous spaces such as compactly and non-compactly causal symmetric spaces and causal flag manifolds.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Orateur: Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

14:30 → 16:00 Problem session

This session will be devoted to exercises and complements related to the mini-course "Algebraic Quantum Field Theory and causal homogeneous spaces" by K.-H. Neeb.

Orateur: Tobias Simon (Friedrich-Alexander Universität Erlangen-Nürnberg (FAU))

16:00 → 17:00 Standard subspaces and distribution vectors

Standard subspaces in a Hilbert space are closely related to the modular and Tomita--Takesaki theory theory, two cornerstones in Algebraic Quantum Field Theory (AQFT). In this talk we start with a short overview of the theory of standard subspaces and the connection to the Tomita--Takesaki theory. We will then discuss Euler elements and causal symmetric spaces and then connect those two subjects via representation theory, analytic continuation to the crown and then limit processes to realize standard subspaces in spaces of distribution vectors.

Orateur: Gestur Olafsson (Louisiana State University)

mer. 19 mars

10:00 → 10:45 Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 3/4 (pt.1)

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$.

Lecture 3: Analytic continuation of orbit maps and crown domains.

The construction of interesting nets of real subspaces rests on the existence of holomorphic extension of orbit maps in unitary representations. For semisimple groups, complex crowns of Riemannian symmetric spaces $G/K$ provide a natural context for this extension process. We explain how this can be set up on general Lie groups whose Lie algebra contains an Euler element.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Orateur: Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

11:00 → 11:45 Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 3/4 (pt.2)

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$.

Lecture 3: Analytic continuation of orbit maps and crown domains.

The construction of interesting nets of real subspaces rests on the existence of holomorphic extension of orbit maps in unitary representations. For semisimple groups, complex crowns of Riemannian symmetric spaces $G/K$ provide a natural context for this extension process. We explain how this can be set up on general Lie groups whose Lie algebra contains an Euler element.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Orateur: Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

14:30 → 16:00 Problem session

This session will be devoted to exercises and complements related to the mini-course "Algebraic Quantum Field Theory and causal homogeneous spaces" by K.-H. Neeb.

Orateur: Tobias Simon (Friedrich-Alexander Universität Erlangen-Nürnberg (FAU))

ven. 21 mars

10:00 → 10:45 Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 4/4 (pt.1)

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$.

Lecture 4: Constructing nets of real subspaces.

Finally, we arrive at rather general characterizations of unitary representations and homogeneous spaces for which a rich supply of nets exists. Many classification results are still open and more bridges to Physics have to be built, but the overall structure of the theory takes shape.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Orateur: Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

11:00 → 11:45 Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 4/4 (pt.2)

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$.

Lecture 4: Constructing nets of real subspaces.

Finally, we arrive at rather general characterizations of unitary representations and homogeneous spaces for which a rich supply of nets exists. Many classification results are still open and more bridges to Physics have to be built, but the overall structure of the theory takes shape.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Orateur: Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

lun. 24 mars

09:00 → 12:00 Workshop: Analysis on homogeneous spaces and operator algebras

See: https://indico.math.cnrs.fr/event/10858/

14:00 → 16:30 Workshop: Analysis on homogeneous spaces and operator algebras

See: https://indico.math.cnrs.fr/event/10858/

mar. 25 mars

09:30 → 12:00 Workshop: Analysis on homogeneous spaces and operator algebras

See: https://indico.math.cnrs.fr/event/10858/

14:00 → 16:30 Workshop: Analysis on homogeneous spaces and operator algebras

See: https://indico.math.cnrs.fr/event/10858/

mer. 26 mars

09:30 → 12:00 Workshop: Analysis on homogeneous spaces and operator algebras

See: https://indico.math.cnrs.fr/event/10858/

jeu. 27 mars

09:30 → 12:00 Workshop: Analysis on homogeneous spaces and operator algebras

See: https://indico.math.cnrs.fr/event/10858/

14:00 → 16:30 Workshop: Analysis on homogeneous spaces and operator algebras

See: https://indico.math.cnrs.fr/event/10858/

16:30 → 17:30 Lecture by Jeffrey Adams: The Unitary Dual — Jeffrey Adams

The Unitary Dual

ven. 28 mars

09:30 → 12:00 Workshop: Analysis on homogeneous spaces and operator algebras

See: https://indico.math.cnrs.fr/event/10858/

14:00 → 17:00 No talks, but the Bourbaki seminar might be of interest

The workshop will be over, but some of the participants may like to attend talks in the Bourbaki seminar on Friday afternoon and on Saturday. For the schedule and abstracts see:

https://www.bourbaki.fr