Maxime Gheysens

Research

My research interests lie in geometric group theory. More precisely, I work in amenability, metric spaces of nonpositive curvature, and bounded cohomology. I'm also interested in the structure theory of locally compact (totally disconnected) groups, property (T), and orbit equivalence. I am doing my PhD under the supervision of Nicolas Monod.

Representing groups against all odds (my thesis)
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We investigate how probability tools can be useful to study representations of nonamenable groups. A suitable notion of 'probabilistic subgroup' is proposed for locally compact groups, and is valuable to induction of representations. Nonamenable groups admit nonabelian free subgroups in that measure-theoretical sense. Consequences for affine actions and for unitarizability are then drawn. In particular, we obtain a new characterization of amenability via some affine actions on Hilbert spaces.

Along the way, various fixed-point properties for groups are studied. We also give a survey of several useful facts about group representations on Banach spaces, continuity of group actions, compactness of convex hulls in locally convex spaces, and measurability pathologies in Banach spaces.

Keywords: amenability, group representation, fixed-point property, von Neumann problem, Dixmier problem, induced representation, tychomorphism, Krein space.
The scale function and tidy subgroups (expository paper)
with A. Brehm, A. Le Boudec, and R. Rollin
arXiv | ShowHide abstract
This exposition article arose from two talks given during the Oberwolfach Arbeitsgemeinschaft on Totally Disconnected Groups in October 2014.

This is an introduction to the structure theory of totally disconnected locally compact groups initiated by Willis in 1994. The two main tools in this theory are the scale function and tidy subgroups, for which we present several properties and examples.

As an illustration of this theory, we give a proof of the fact that the set of periodic elements in a totally disconnected locally compact group is always closed, and that such a group cannot have ergodic automorphisms as soon as it is non-compact.
Fixed points for bounded orbits in Hilbert spaces
with N. Monod
arXiv | ShowHide abstract
Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact σ-compact groups (e.g. countable groups).

Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau–Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for locally compact groups.

We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.

Annales scientifiques de l'École normale supérieure, vol. 50 (2017), p. 131–156.

Talks

Séminaire de Géométrie, Groupes et Dynamiques (ENS de Lyon, France), June 2017
Seminar Geometrie (TU Dresden, Germany), January 2017
Séminaire de théorie des groupes (UCL, Belgium), January 2017
Séminaire « Groupes et Géométrie » (Université de Genève, Switzerland), November 2016
Geometry Seminar (ETHZ, Switzerland), October 2016
Journées d'analyse fonctionnelle Besançon–Neuchâtel (Université de Franche-Comté, France), May 2016
Séminaire de Géométrie Différentielle (Université de Lorraine, France), May 2016
Geometry and Analysis on Groups Research Seminar (Universität Wien, Austria), January 2016
Oberseminar C*-Algebren (WWU Münster, Germany), January 2016
Groups and Analysis Seminar (Université de Neuchâtel, Switzerland), October 2015
Ergodic and Geometric Group Theory Seminar (EPFL, Switzerland), May 2015

Enseignement

Assistanat pour les cours suivants :

Analyse II (printemps 2013 et 2014),
Géométrie et groupes (automne 2013),
Introduction aux variétés différentiables (automne 2014),
Analysis on Groups (printemps 2015),
Logique mathématique (automne 2015),
Analysis on Groups (printemps 2016),
Analysis on Groups (printemps 2017).

Encadrement d'étudiants pour divers projets de semestre (voir la page des projets de semestre de la chaire EGG).

Divers

Textes et exposés prédoctoraux (disponibles sur demande)

Actions de groupes sur des espaces de Banach (mémoire de master sous la direction de Nicolas Monod, septembre 2012).
La propriété (T) de Kazhdan et les immeubles triangulaires (en collaboration avec Nicolas Garrel, sous la direction de Frédéric Paulin, juin 2011).
Classification des groupes de Coxeter finis (sous la direction de Dimitri Leemans, novembre 2009).
Méthode probabiliste en théorie des graphes (exposé pour la Brussels Summer School of Mathematics, août 2009).

Responsabilités administratives

Édition du rapport de l'Arbeitsgemeinschaft Totally disconnected groups d'Oberwolfach (octobre 2014).

Organisation d'un groupe de travail sur la structure des groupes simples totalement discontinus (printemps 2014).

Organisation du quatorzième colloque des doctorants en mathématique de Suisse occidentale (février 2015).
Délégué des doctorants mathématiciens de l'EPFL auprès du Programme doctoral suisse en mathématiques (depuis mars 2014).

Contact