French version

Homepage of Owen Garnier Temporary research and teaching assistant at the University of Amiens (Université de Picardie Jules Verne)

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Welcome !

I am currently a Temporary research and teaching assistant at the Université de Picardie Jules Verne. I defended my PhD thesis at the LAMFA in 2024, under the supervision of Ivan Marin.

Here you can find the material for my courses, my publications and the notes of my talks.
Besides, I passed the agrégation externe de mathématiques in 2020 and you will also find here my lesson plans and "développements" ! (in french only)

Description of my work

My research interests lie in combinatorial group theory, more specifically in Garside theory and its application in the study of complex braid groups.

If \(V\) is a finite dimensional complex vector space, and \(W\) is a finite subgroup of \(\mathrm{GL}(V)\) generated by reflections, the braid group \(B(W)\) attached to \(W\) is defined as the fundamental group of the space \(X/W\) of regular orbits attached to \(W\). Although complex braid groups have a topological definition, they can be studied via Garside theory.

A Garside group structure on a group \(G\) is the data of a triple \((G,M,\Delta)\), where \(M\subset G\) is a submonoid and \(\Delta\in M\), both satisfying several combinatorial assumptions. Such a structure gives many informations on the group \(G\) (conjugacy problem, presentation, homology computations...). Almost every complex braid group can be endowed with one (or several) Garside group structures.

My works are aimed in particular at the complex reflection group \(G_{31}\) and its braid group. This group has no known Garside group structure, however it admits a Garside groupoid structure. A Garside groupoid structure is a triple \((\mathcal{G},\mathcal{C},\Delta)\), where \(\mathcal{G}\) is a groupoid, \(\mathcal{C}\subset \mathcal{G}\) is a subcategory and \(\Delta:\mathrm{Ob}(\mathcal{C})\to \mathcal{C}\) is a map. Here again, both satisfy several combinatorial assumptions. As in the case of Garside groups, the study of the Garside groupoid attached to \(B(G_{31})\) gives many informations on this group.