Research in Mathematics

My field of research is Combinatorial and geometric Group and Semigroup theory
I  work on the following topics: decision problems in groups and monoids , like the word problem , the conjugacy problem, the membership problem; the existence of rewriting systems, left-orderability of groups and other related questions with a particular interest in the following families of groups:  knot groups, braid groups, Artin groups, Garside groups, structure groups related to the Yang-Baxter equation, Bieberbach groups..



What can be said about Combinatorial group theory ?
Combinatorial group theory is now a very vast domain in group theory, with connections to topology, logic, graph theory, computer science and others. Roughly speaking, combinatorial group theory  studies groups defined by a presentation,  that is by means of generators and relations.  Its  origins  can  be  traced  back  to  the  middle  of  the  19th  century; it emerged with   the introduction of the fundamental group  of a manifold by Poincare in 1895.  In 1908, Tietze proved that the  fundamental group of a compact finite dimensional arc-wise connected manifold is a finitely presented group and this raised many questions about   finitely presented groups. In particular, a few years later, Max Dehn formulated three fundamental well-known decision problems: the word problem, the conjugacy problem and the isomorphism problem. Despite the apparent simplicity of  the formulation of these problems,  they are unsolvable in general. In view of  the extraordinary variety of  existing groups, (there are uncountably many finitely generated groups and countably many finitely presented groups),  this led to define  particular 
 classes of groups and to try to answer  for them Dehn's questions. As time passed, a very wide range of  questions (group-theoretic, algorithmic, geometric..) arose and with them the development of a large  variety  of techniques and methods, so that  the boundaries of the  combinatorial theory field have been widened.