Grant: Discovery Grants Program - Individual

Although my research program concerns the study of some abstract algebraic structures (Coxeter groups), it admits ramifications ranging from theoretical physics to statistics, throughout various applications in other fields of mathematics. Coxeter groups appear naturally as a set of transformations (symmetry) of regular polyhedrons, and therefore are playing an important role in crystallography. They can also be found virtually in almost all areas of mathematics. For example in relation to simple Lie algebras and Kac-Moody algebras, or in some aspects of refined study of Euclidean geometry and hyperbolic geometry. The originality of my approach is to exploit the combinatorial properties of these groups as the key for a better understanding of their related structures. In particular, the aim of my research is the creation of new combinatorial tools arising from the study of Coxeter groups to refine the knowledge we have of their associated structures. More specifically, the purpose is to have a clearer understanding of the concept of Descent algebras, Combinatorial Hopf algebras and Generalized Associahedra. For instance, generalized associahedra are fundamental geometric objects in the study of cluster algebras, whose ramifications extend to physics, thermodynamics and statistics. ) ) )