Coxeter groups and related structures

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My primary area of research is algebraic combinatorics. It is a highly active area of mathematics, with many connections to algebraic geometry, convex geometry, representation theory, topology, mathematical physics and statistical mechanics, among many others. I am more precisely interested on the combinatorial and geometrical aspects of the study of Coxeter (mirror reflection) groups and their related structures. Coxeter groups appear in very many domains of mathematics, for instance, as symmetry groups of regular polytopes, as Weyl groups of semi-simple Lie algebras and Kac-Moody algebras, and as triangle groups in geometry (Euclidean and hyperbolic). Properties of these groups are often key to the understanding of related structures. It is well-established that root systems are fundamental in the theory of Coxeter groups. While finite and affine root systems have been given a lot of attention, almost nothing is known for general infinite root systems. I have uncovered recently, with J.-P.~Labbé (Berlin), V.~Ripoll (UQAM), an exciting new approach that consists in the study of the limit points of roots, opening up a large program of research in several directions, each worthy of independant study. These directions go from applications to Kac-Moody algebras to analogs of generalized associahedra in the infinite case and include generalizations of weak order on root systems. whereas generalized associahedra are fundamental geometric objects in the study of cluster algebras, whose ramifications extend to physics, thermodynamics and statistics. I plan to exploit my research. Another line of research I will pursue relate Coxeter groups with the study of Descent algebras. Descent algebras are key ingredients in an enriched version of the representation theory of finite Coxeter groups. I plan to uncover this enriched structure by exploiting my past work on symmetric groups and hyperoctahedral groups, together with a new idea based on a "type D Hopf algebra".