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 ot 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 — Minicourse: structure of tempered homogeneous spaces (T. Kobayashi)

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

March 3 to 7 — p-adic week

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

• The Bernstein center, by David Renard (4h30)
• Hochschild homology of reductive p-adic groups, by Maarten Solleveld (4h)
• Research talks

March 10 to 14 — Theta correspondence week

Coordinated by Hung Yean Loke and Tomasz Przebinda.

• Local theta correspondence, by Chengbo Zhu (4h30)
• The Howe correspondence for reductive groups over finite fields, by Anne-Marie Aubert (4h)
• Exceptional theta correspondences, by Gordan Savin (4h)
• Research talks

March 17 to 21 — Minicourse: AQFT and causal homogeneous spaces (K.-H. Neeb)

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

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

IMPORTANT WARNING:  Scam / Phishing / SMiShing ! Note that ill-intentioned people may be trying to contact some of participants by email or phone to get money and personal details, by pretending to be part of the staff of the conference center.  IHP and the organizers will NEVER contact you by phone on this issue and will NEVER ask you to pay for accommodation/ board / possible registration fee in advance. 

In particular, do not reply to emails from  operations@conventionsaccommodation.com

Financial support requests are now closed.

Feb11-12_prog.pdf

JulgFest_prog.pdf

Poster-T1-2025 RTNCG.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-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

lundi 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)

mardi 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)

mercredi 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)

jeudi 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

vendredi 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)

lundi 20 janvier

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

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

mardi 21 janvier

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

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

mercredi 22 janvier

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

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

jeudi 23 janvier

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

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

vendredi 24 janvier

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

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

lundi 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)

mardi 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)

mercredi 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)

jeudi 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)

vendredi 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)

lundi 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)

mardi 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)

mercredi 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)

jeudi 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)

vendredi 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)

mardi 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)

mercredi 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)

jeudi 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)

lundi 17 février

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

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

mardi 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 Title TBA

Joint IHP - AIM RTNCG lecture

Abstract: Coming soon.

Orateur: Nicolas Prudhon (université de Lorraine)

mercredi 19 février

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

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

vendredi 21 février

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

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

lundi 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

mardi 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/

mercredi 26 février

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

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

jeudi 27 février

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

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

15:00 → 17:30 Workshop: Tempered representations and K-theory

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

vendredi 28 février

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

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