We show that up to a null set, every infinite measure-preserving action of a locally compact Polish group can be turned into a continuous measure-preserving action on a locally compact Polish space where the underlying measure is Radon. We also investigate the distinction between spatial and boolean actions in the infinite measure-preserving setup. We finally obtain a streamlined proof of a recent result of Avraham- Re’em and Roy: Lévy groups cannot admit nontrivial continuous measure-preserving actions on Polish spaces when the measure is locally finite.
Research
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Spatial models for boolean actions in the infinite measure-preserving setup (with François Le Maître),
submitted (2024). arXiv, pdf.
ABSTRACT.
Talks
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12/02/2025: LAREMA PhD seminar, Université d'Angers, Angers.
ABSTRACTDans cet exposé je parlerai d'un invariant d'équivalence orbitale introduit par H. Dye en 1959, le groupe plein. Dans un premier temps je définirai cet objet et je donnerai quelques exemples, puis j'énoncerai le théorème principal qui en motive l'étude, le théorème de reconstruction. Ce théorème encourage l'étude de ces groupes pleins en tant que groupes topologiques, et ainsi dans un deuxième temps je parlerai de certaines topologies naturelles sur les groupes pleins, le théorème principal de cette partie affirmant que dans certains cas on peut toujours trouver une topologie Polonaise. Enfin, si le temps le permet, je discuterai de certaines propriétés topologiques, en insistant sur les différences avec le cas d'une mesure de probabilité.
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10/01/2025: Groupe de travail de théorie ergodique, Université Paris Cité, Paris.
ABSTRACTDye's reconstruction theorem states that any abstract isomorphism between two ergodic full groups on a standard probability space is actually a conjugation by some measure-preserving bijection. This is useful for instance for proving that full groups are complete invariants of orbit equivalence. When working with infinite measure or with non-singular bijections however, we need a more general version of the theorem, which was later proved by Fremlin in the language of Boolean algebras. I will give some details about his proof, which relies on an algebraic description of the involutions in the full group, and will work specifically in the context of measure algebras. See Notes.
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13/12/2024: Arbre de Noël 2024 en Géométrie Non Commutative, Orléans, France.
ABSTRACTAny measure-preserving Borel action of a Polish group on a standard measured space -called a spatial action- gives a measure-preserving boolean action, an action on the Borel subsets up to measure zero. Conversely, a natural problem is the following: given a boolean action, does it come from a spatial action? Glasner, Tsirelson and Weiss gave a complete answer for finite measures. Moreover, when a spatial action exists, a theorem of Becker and Kechris implies that we can always take it to be a continuous action on a compact space. For the case of infinite measures however, we need to take into account the interplay between measure and topology, and to this end we aim to obtain a continuous action on a locally compact Polish space endowed with a Radon measure. In a joint work with François Le Maître, we show that it is possible to obtain such a model for actions of lcsc groups.
About the article: Spatial models for boolean actions in the infinite measure-preserving setup. See Research. -
11/04/2024: Séminaire d'Algèbres d'Opérateurs, IMJ-PRG, Paris.
ABSTRACTAny measure-preserving Borel action of a Polish group on a standard measured space -called a spatial action- gives a measure-preserving boolean action: an action on the Borel subsets up to measure zero. Conversely, a natural problem is the following: given a boolean action, does it come from a spatial action? Glasner, Tsirelson and Weiss gave a complete answer for finite measures. Moreover, when a spatial action exists, a theorem of Becker and Kechris implies that we can always take it to be a continuous action on a compact space. For the case of infinite measures however, we need to take into account the interplay between measure and topology, and to this end we aim to obtain a continuous action on a locally compact Polish space endowed with a Radon measure. In a joint work with François Le Maître, we show that it is possible to obtain such a model for actions of locally compact groups, while spatial actions of a certain class of large groups can only be trivial.
About the article: Spatial models for boolean actions in the infinite measure-preserving setup. See Research. -
08/04/2024: Séminaire Groupes et actions, Université Paris-Saclay, Orsay.
ABSTRACTAny measure-preserving Borel action of a Polish group on a standard measured space -called a spatial action- gives a measure-preserving boolean action, an action on the Borel subsets up to measure zero. Conversely, a natural problem is the following: given a boolean action, does it come from a spatial action? Glasner, Tsirelson and Weiss gave a complete answer for finite measures. Moreover, when a spatial action exists, a theorem of Becker and Kechris implies that we can always take it to be a continuous action on a compact space. For the case of infinite measures however, we need to take into account the interplay between measure and topology, and to this end we aim to obtain a continuous action on a locally compact Polish space endowed with a Radon measure. In a joint work with François Le Maître, we show that it is possible to obtain such a model for actions of locally compact groups, while spatial actions of a certain class of large groups can only be trivial.
About the article: Spatial models for boolean actions in the infinite measure-preserving setup. See Research. -
30/05/2023: Groupe de travail de théorie ergodique, Université Paris Cité, Paris.
ABSTRACTAbout the proof of Belinskaya's theorem. See Notes.
Notes
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Some notes about Fremlin's generalization of Dye's reconstruction theorem, for the ergodic theory working group.
ABSTRACTIn this note we provide context and give a demonstration of a generalization of Dye’s reconstruction theorem, proved by Fremlin, following Eigen’s arguments. The theorem is stated for the very general setting of Dedekind complete Boolean algebras and their automorphism groups, but we will specifically work with measure algebras of standard spaces.
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Some handwritten notes about Belinskaya's theorem (in french), for the ergodic theory working group.
ABSTRACTThese notes provide a detailed proof of Belinskaya's theorem (two ergodic pmp bijections of a standard probability space that are orbit equivalent are flip conjugate as soon as one of the cocycles is integrabe), due to Katznelson, and published in this article by Alessandro Carderi, Matthieu Joseph, François Le Maître and Romain Tessera.
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My Master thesis on Property (T) for measured groupoids (in french), supervised by François Le Maître.
ABSTRACTThis master thesis is based on an article from Claire Anantharaman-Delaroche, and more specifically about a theorem on Kazhdan's property (T) for measured groupoids. The necessary background about groupoids, measure disintegration, unitary representations and affine Hilbert spaces is introduced, then the main theorem is stated and proved. A direct consequence is the following: if G is a countable group acting in a free, pmp and ergodic manner on a standard probability space, then the associated equivalence relation has property (T) if and only if G has property (T). Some comments are made at the end about an extension of the theorem to the locally compact setting.
Teaching
- 2023- Colles (Regular oral examinations) for students of ECG1 Applied Mathematics, Lycée Alexandre Dumas, Saint-Cloud.
- 2021- High school level Mathematics courses, Lycée Alexandre Dumas, Saint-Cloud.
- 2020-2021 High school level Mathematics courses, Lycée Alain, Le Vésinet.
- 2019-2020 Middle school level Mathematics courses, Collège Jean Rostand, Nîmes.
- 2018 Agrégation externe de Mathématiques.