Arbre de Noël 2019 du GDR GNC 17-19 décembre 2019 Paris 7, bâtiment Sophie Germain, amphi Turing

À propos

Cette année la rencontre annuelle du GDR de Géométrie Non Commutative aura lieu à l'IMJ-PRG.

Nous aurons des exposés pléniers de Claire Anantharaman-Delaroche, Giovanni Landi, Adrien Le Boudec, Yves Stalder, Stéphane Vassout et Roland Vergnioux.

Nous encourageons les jeunes chercheuses et chercheurs à présenter leurs travaux, et pouvons financer leur venue et leur hébergement.

Programme

All talks take place at amphi Turing, in the Sophie Germain building.

	Mardi	Mercredi	Jeudi
09:00 – 10:00	Roland Vergnioux	Claire Anantharaman	Ignacio Vergara
10:00 – 10:30	Coffee break
10:30 – 11:30	Hua Wang	Simeng Wang	Xiao Xiong
11:30 – 12:30	Isabelle Baraquin	Nóra Szőke	Giovanni Landi
12:30 – 14:30	Lunch break
14:30 – 15:30	Stéphane Vassout	Adrien Le Boudec
15:30 – 16:00	Coffee break
16:00 – 17:00	Omar Mohsen	Yves Stalder

• Contenance faible vs moyennabilité pour les groupoïdes et les actions de groupes
  Claire Anantharaman-Delaroche (Université d'Orléans)

  On dit qu’un groupoïde ou qu’une action de groupe sur une C*-algèbre a la propriété de contenance faible si les C*-algèbres pleine et réduite associées coïncident. Pour un groupe il est bien connu que cette propriété est équivalente à la moyennabilité. Comprendre dans quelle mesure la contenance faible implique la moyennabilité en général est une question subtile que l’on va discuter dans cet exposé.

• De Finetti Theorems
  Isabelle Baraquin (Université de Lorraine)

  In this talk, we will present de Finetti Theorem and some similar results in noncommutative probability. After looking at the quantum case, we will study a de Finetti theorem in the dual unitary group.

• Solutions to the quantum Yang–Baxter equation and related deformations
  Giovanni Landi (Università di Trieste)

  We present natural families of coordinate algebras of noncommutative Euclidean spaces and noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the quantum Yang–Baxter equation. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have spherical manifolds, and noncommutative quaternionic planes as well as noncommutative quaternionic tori. On these there is an action of the classical quaternionic torus SU(2) x SU(2) in parallel with the action of the torus U(1) x U(1) on a complex noncommutative torus.

• Transitivity degree and non-free dynamical actions
  Adrien Le Boudec (ENS Lyon)

  The transitivity degree of a group G is an invariant of G associated to representations of G as a permutation group: this is the largest integer k such that G admits a faithful k-transitive action; and if there is no such integer then the transitivity degree is infinite. For several classes of groups coming from geometric group theory, this invariant is known to be infinite. Recently Gelander--Glasner--Soifer showed that the same holds for Zariski dense subgroups of SL(2,K), where K is a local field.
  In the talk we will discuss the situation of certain groups coming from "dynamical group theory". For example for groups having a minimal and non-free action on the circle, and groups having a minimal and non-free action on the boundary of a tree, we will see that the transitivity degree can be explicitly computed, and is determined by the action of G on its orbits on the circle or on the boundary of the tree. This is joint work with Nicolas Matte Bon.

• Index theorem for inhomogeneous differential operators
  Omar Mohsen (Universität Münster)

  I will present a cohomological index formula for the analytic index of differential operators satisfying the Rockland condition (so hypoelliptic but not necessarily elliptic). This generalises previous work by Erik Van-Erp.

• More highly transitive groups among groups acting on trees
  Yves Stalder (Université Blaise Pascal)

  Highly transitive groups, i.e. groups admitting an embedding in Sym(N) with dense image, form a wide class of groups. For instance, Hull and Osin proved that it contains all countable acylindrically hyperbolic groups with trivial finite radical. In this talk, I will present a theorem showing that many groups acting on trees are highly transitive. On the one hand, this theorem applies to some non acylindrically hyperbolic groups. On the other hand, it is sharp because of results by A. Le Boudec and N. Matte Bon. The case of Baumslag-Solitar groups will be presented. This is joint work with P. Fima, F. Le Maître and S. Moon.

• A Tits alternative for topological full groups
  Nóra Szőke (Université Grenoble Alpes)

  I will present a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, the topological full group of a minimal action of a virtually cyclic group is amenable. This is a generalization of the result of Juschenko and Monod for Z-actions. On the other hand, when a finitely generated group G is not virtually cyclic, then we can construct a minimal free action of G on a Cantor space such that the topological full group contains a non-abelian free group.

• Fourier Integral Operators on Lie Groupoids
  Stéphane Vassout (Université de Paris)

  Based on our work with Jean-Marie Lescure, I will explain the construction of Fourier Integral Operators recently developed in the framework of Lie groupoids. This calculus not only contains the original calculus for smooth compact manifolds, but also fits with many different generalizations of this calculus developped in various geometric situations. This calculus is in particular based on the Coste-Dazord-Weinstein sypmplectic Lie groupoid structure of the cotangent space T*G of a given Lie groupoid G.

• Positive definite radial kernels on homogeneous trees and products
  Ignacio Vergara (IMPAN)

  A classical result in harmonic analysis on homogeneous trees characterises positive definite radial kernels in terms of Borel measures on the interval [-1,1]. I will discuss a new proof of this result, which relies on a characterisation of radial Schur multipliers by Haagerup--Steenstrup--Szwarc and a variation of the Hamburger moment problem. Moreover, this new method allows one to extend the result to finite products of trees.

• Fürstenberg boundary for discrete quantum groups
  Roland Vergnioux (Université de Caen)

  I will report on a recent joint work with Kalantar, Kasprzak and Skalski where we introduce a notion of noncommutative Fürstenberg boundary for discrete quantum groups. In the classical case, faithfulness of the corresponding action is equivalent to triviality of the amenable radical and to the unique trace property. I will discuss the quantum generalization of these results and the case of orthogonal free quantum groups.

• Classification of irreducible representations of semi-direct products of a compact quantum group with a finite group
  Hua Wang (Université de Paris)

  It is trivial to classify irreducible representations of a direct product of groups in terms of the representations of the factor groups. But once we replace the direct product by the much more ubiquitous semi-direct product, the situtation quickly becomes complicated. In the 20th century, G. Mackey developed a powerful machinery, aka. Mackey’s analysis, to treat such questions in the context of locally compact groups. In this talk, I will present a quantum analogue of Mackey’s analysis, which is inspired by the ideas of rigid C*-tensor categories, in the context of semi-direct products of a compact quantum group with a finite group. In particular, I will mainly focus on classifying all irreducible representations of such quantum groups. If time permits, I will mention the more delicate topic of fusion rules of these quantum groups, and perhaps some interesting applications.

• Convergence presque partout des séries de Fourier non commutatives
  Simeng Wang (Université Paris-Sud)

  Il est bien connu qu'un certain nombre de propriétés d'approximation de groupes s’interprètent comme des méthodes de sommabilité des séries de Fourier dans les espaces L^p non commutatifs associées. Dans cet exposé je présenterai les résultats plus raffinés dans ce cadre, à savoir la sommabilité « presque partout » de ces séries. Nous prouvons que pour tout groupe dénombrable discret moyennable, il existe une suite de fonctions de type positif à support fini, telle que les multiplicateurs de Fourier associés sur les espaces L^p non commutatifs satisfassent à la convergence presque uniforme vers l’identité au sens de Lance pour tout p>1. De façon similaire, nous obtenons les résultats pour nombreux groupes ou groupes quantiques avec la propriété de Haagerup ou la moyennabilité faible. En particulier, nos méthodes entraînent les nouveaux résultats et approches de l’analyse classique sur les groupes compacts à l’aide de la géométrie non commutative. D'autre part, nos arguments déduisent également les inégalités maximales de Hardy-Littlewood non commutatives avec une borne indépendante de la dimension.

• Quantum differentiability on quantum tori and quantum euclidean spaces
  Xiao Xiong (Harbin Institute of Technology)

  The core ingredients of the quantized calculus, introduced by A. Connes, are a separable Hilbert space H, a unitary self-adjoint operator F on H and a C*-algebra A represented on H such that for all a ∈ A the commutator [F,a] is a compact operator on H. Then the quantized differential of a ∈ A is defined to be the operator da = i [F,a]. We provide a full characterization of quantum differentiability in the sense of Connes on quantum tori T_θ^d and quantum euclidean spaces R_θ . We also prove a quantum integration formula which differs substantially from the commutative case.
  Based on joint work with Edward McDonald and Fedor Sukochev.

Organisation

• Pierre Fima : pierre.fima[arobase]imj-prg.fr
• François Le Maître : francois.le-maitre[arobase]imj-prg.fr
• Maria Paula Gomez : maria.gomez[arobase]math.u-psud.fr

Détails

The meeting will take place in Paris 7, in the building Sophie-Germain, the room is amphi Turing.

• Lunch breaks take place at restaurant Buffon, located at 17 rue Hélène Brion.
• On Wednesday evening, we will have a conference dinner. It will take place at restaurant la Touraine, located at 39 rue de Croulebarbe.

Hébergement et voyage

• Nous pouvons financer la venue et l'hébergement de jeunes chercheurs et chercheuses, vous pouvez envoyer une demande à Pierre Fima jusqu'au 18 octobre.
• Nous recommandons aux personnnes dont nous ne finançons pas le séjour de réserver un hôtel le plus tôt possible. Voici quelques possibilités, certaines étant à une petite distance à métro de Paris 7:
  • Hotel Ibis Budget Porte d'Italie
  • Lodge In
  • Hotel Coypel
  • Timhotel
  • Hotel Ibis Paris Italie Tolbiac 13eme
  • Jack's Hotel

Financements

Cette rencontre est financée par le GDR Géométrie non commutative, l'IMJ-PRG (également l'équipe d'algèbres d'opérateurs), l'ANR SingStar et l' UFR de Maths de Paris 7.