Dans cette thèse, on étudie des marches aléatoires induites par des actions de groupes probabilisés sur des espaces de courbure négative ou nulle. On décrit en particulier le cas d'actions sur des espaces CAT(0) admettant des éléments contractants, c'est-à-dire dont les propriétés imitent ceux des isométries loxodromiques dans les espaces Gromov-hy...
Given a group Γ, its Bohr compactification Bohr(Γ) and its profinite completion Prof(Γ) are compact groups naturally associated to Γ; moreover, Prof(Γ) can be identified with the quotient of Bohr(Γ) by its connected component Bohr(Γ)_0. We study the structure of Bohr(Γ) for an arithmetic subgroup Γ of an algebraic group G over Q. When G is unipoten...
The aim of the poster is to showcase the interplay between group theory, algebraic topology and combinatorics on words. A result that allows to display this is the return theorem by Berté et al. in 2015. The poster will contain an introduction to fundamental groups of graphs, dendric words as well as a new result concerning return groups of eventua...
Since 2015, dendric shifts (a generalisation of Sturmian words) have been widely studied. One of the results concerning these shift spaces is the return theorem. It describes the groups generated by the return words of a dendric shift. The proof uses the fundamental group of the Rauzy graph of the shift space. Later, eventually dendric shifts were ...
L'objectif de cet article est de définir un nouvel espace géométrique qui étend la notion de vecteur à des objets géométriques que nous nommerons secteur (portion de surface sous forme de parallélogramme). Nous en extrairons 3 propositions de structures algébriques qui étendent les notions de groupe, corps et espace vectoriel. Ces structures seront...
Convolutional neural networks have proven to be the deep learning model that performs best on regularly structured datasets like images or sounds. However, they cannot be applied on datasets with an irregular structure (e.g. sensor networks, citation networks, MRIs). In this thesis, we develop an algebraic theory of convolutions on irregular domain...
À l'échelle nanométrique, les particules métalliques présentent de nouvelles propriétés optiques liées au phénomène de résonance plasmon de surface. Une résonance plasmon est une oscillation collective et cohérente des électrons de conduction à la surface d'une nanoparticule métallique sous l'influence d'un champ électromagnétique extérieur. La lon...
The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their i...
Published in International Journal of Research in Undergraduate Mathematics Education
What is questioned and what is taken for granted when carrying out research into the teaching of a given mathematical topic such as Group Theory? This paper presents two different questioning procedures using the methodological tools provided by the Anthropological Theory of the Didactic (ATD). The first one, leading to an undergraduate instruction...
Shift spaces are sets of colorings of a group which avoid a set of forbidden patterns and are endowed with a shift action. These spaces appear naturally as discrete versions of dynamical systems: they are obtained by partitioning the phase space and mapping each element into the sequence of partitions visited by its orbit.Severa! breakthroughs in t...
