An IRS on a locally compact group is a conjugacy-invariant probability measure on the space of its subgroups.
We are interested in those IRS that give full measure to the set of compact subgroups.
This talk is about what we know about those, how it connects to the structure theory of locally compact groups,
and what we would still like to figure out.
Joint with Tal Cohen, Helge Glöckner and Gil Goffer.
Dans cet exposé, nous parlerons de 2 modèles classiques de croissance aléatoire en théorie des probabilités qui sont la Percolation de Premier Passage et le Processus de Contact.
Ces deux modèles interviennent dans la modélisation de la propagation de feux de forêt, d'épidémies ou de fluide dans un mieux poreux.
On considère un graphe G=(V,E) et on imagine une infection affectant les sites de V et se propageant aléatoirement via les arêtes de E.
On s’intéresse à l'évolution de l'ensemble aléatoire des sites infectés à un certain temps (ou bien avant un certain temps).
Nous parlerons de théorème ergodique sous additif, de forme asymptotique et éventuellement de mesure invariante.
Nous citerons Kesten, Harris, Kingman et bien d'autres.
La plupart des résultats porteront sur le réseau Zd mais n'excluons pas quelques digressions sur d'autres graphes.
It is a classical result of Norbert Wiener from the 1930’s, referred to as “Wiener’s Tauberian theorem”,
that a function f in L^1(R^n) generates a dense ideal if and only if its Fourier transform vanishes nowhere.
Then, given a general locally compact group G, one can ask whether the Fourier transform on L^1(G) also satisfies this property.
In the case that this property holds for L^1(G), G is called a “Wiener group”.
It was a classical question in Banach algebra theory during the 20th century to determine which groups are Wiener.
It is a celebrated result in the area that compactly generated groups of polynomial growth and nilpotent groups are all Wiener groups.
On the other hand, it is generally difficult to show that a group is not Wiener,
and essentially the only known class of non-Wiener groups are connected semisimple Lie groups.
In this talk I will discuss recent work where we show that many non-amenable totally disconnected locally compact groups are not Wiener,
including reductive algebraic groups over non-archimedean local fields.
The proofs make extensive use of representations of the given groups on their Furstenberg boundary.
I will give an introduction to each of these topics during the talk.
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 Tπ 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 Tπ, 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.
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 Tπ, 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.
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.
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.
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.
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.
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.
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.
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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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.