Combinatorics of coxeter groups and relations with their relative structures

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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.