Groupe de travail de théorie ergodique

Coorganisé avec Matthieu Joseph. Le groupe de travail se réunit tous les vendredis à 10h en salle 1013 du bâtiment Sophie Germain.

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Séances précédentes

• 20 mars : Actions Gaussiennes et propriété de Foiaș et Strătilă pour les actions de O(H) (Emmanuel Roy). Voici l'enregistrement, les notes seront mises à disposition prochainement, voici également l'abstract.
  Les espaces Gaussiens sont des sous-espaces L² d’un espace de probabilité ayant la propriété remarquable de n’être constitués que de variables Gaussiennes. Ces sous-espaces sont liés aux actions dites Gaussiennes qui permettent de définir une action p.m.p. d’un groupe G à partir d’une de ses représentations orthogonales. On obtient ainsi une action T_π de O(H), le groupe orthogonal d’un espace de Hilbert réel séparable de dimension infinie, à partir de sa représentation canonique π, l’identité sur O(H).
  Dans cet exposé, nous montrerons que cette action satisfait la propriété de rigidité suivante: Si une action p.m.p. de O(H) possède une représentation de Koopman équivalente à celle de T_π, alors les deux actions sont isomorphes. Ce résultat est l’analogue pour O(H) d’un théorème dû à Foiaș et Strătilă pour des actions de Z, où les mesures spectrales sont remplacées ici par des représentations.
  Travail en collaboration avec Tomás Ibarlucía.
• 20 février : On dense orbits in the Kechris spaces of pmp equivalence relations II (François Le Maître). Voici les notes de l'exposé (en français), l'enregistrement ainsi que l'abstract.
  The space of subequivalence relations, or Kechris space, of a given non-singular equivalence relation can be endowed with a natural Polish topology whose study was initiated by Kechris. He rose the following natural question: when is the action of the full group (or the automorphism group) of a given ergodic non-singular equivalence relation on its Kechris space topologically transitive? When does this action have meager orbits? I will explain why the full group of the ergodic hyperfinite pmp equivalence relation acts topologically transitively with meager orbits on its Kechris space, and discuss the many remaining questions.
• 13 février : On dense orbits in the Kechris spaces of pmp equivalence relations I (François Le Maître). Voici les notes de l'exposé (en français), l'enregistrement et l'abstract.
  The space of subequivalence relations, or Kechris space, of a given non-singular equivalence relation can be endowed with a natural Polish topology whose study was initiated by Kechris. He rose the following natural question: when is the action of the full group (or the automorphism group) of a given ergodic non-singular equivalence relation on its Kechris space topologically transitive? When does this action have meager orbits? I will explain why the full group of the ergodic hyperfinite pmp equivalence relation acts topologically transitively with meager orbits on its Kechris space, and discuss the many remaining questions.
• 30 janvier : Some (more) ergodic properties of Poisson boundaries and applications II (Corentin Le Bars). Voici les notes de l'exposé (en français), l'enregistrement ainsi que l'abstract.
  Poisson boundaries represent the asymptotic significant behaviour of a stochastic process on a group. I will present some of their strong ergodic properties, and show that they naturally appear in some superrigidity problems in higher rank. This will be a sort of continuation of Antoine Derimay's talks, but I will introduce all the concepts.
• 9 janvier : Some (more) ergodic properties of Poisson boundaries and applications I (Corentin Le Bars). Voici les notes de l'exposé (en français), l'enregistrement ainsi que l'abstract.
  Poisson boundaries represent the asymptotic significant behaviour of a stochastic process on a group. I will present some of their strong ergodic properties, and show that they naturally appear in some superrigidity problems in higher rank. This will be a sort of continuation of Antoine Derimay's talks, but I will introduce all the concepts.
• 5 décembre : The Poisson point process from scratch II (Sam Mellick) Voici l'enregistrement, des notes détaillées, et l'abstract.

  The Poisson point process is a fundamental object from stochastic geometry, with many real world applications. I will present this object from the ergodic theoretic perspective, and show in full detail how it is constructed, as well as prove many of its basic properties. Some probability theory knowledge will be assumed (but only a first undergrad probability course, and I will remind everyone of the key tools needed as they come up).

  Whilst there is no particular guiding goal theorem, one should note that the Poisson point process has played an integral role in recent breakthroughs for proving fixed price one for higher rank semisimple Lie groups and products of groups, and this hopefully serves as sufficient motivation.

• 14 novembre : The Poisson point process from scratch I (Sam Mellick) Voici l'enregistrement, des notes détaillées et l'abstract.

  The Poisson point process is a fundamental object from stochastic geometry, with many real world applications. I will present this object from the ergodic theoretic perspective, and show in full detail how it is constructed, as well as prove many of its basic properties. Some probability theory knowledge will be assumed (but only a first undergrad probability course, and I will remind everyone of the key tools needed as they come up).

  Whilst there is no particular guiding goal theorem, one should note that the Poisson point process has played an integral role in recent breakthroughs for proving fixed price one for higher rank semisimple Lie groups and products of groups, and this hopefully serves as sufficient motivation.

• 10 octobre : Random walks and Poisson boundaries in ergodic theory II (Antoine Derimay). Voici l'enregistrement, les notes ainsi que l'abstract.
  To a random walk on a group, one can associate a measured space on which the group acts by measure-class preserving automorphisms, called the Poisson boundary of the walk. This action is the equality case of entropy inequalities, and it satisfies strong ergodic properties and is amenable, which makes it a boundary in the sense of Bader-Furman. As such, a good understanding of this action can lead to various interesting rigidity results.
• 3 octobre : Random walks and Poisson boundaries in ergodic theory I (Antoine Derimay). Voici l'enregistrement, les notes ainsi que l'abstract.
  To a random walk on a group, one can associate a measured space on which the group acts by measure-class preserving automorphisms, called the Poisson boundary of the walk. This action is the equality case of entropy inequalities, and it satisfies strong ergodic properties and is amenable, which makes it a boundary in the sense of Bader-Furman. As such, a good understanding of this action can lead to various interesting rigidity results.

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