Algebraic and geometric combinatorics of Coxeter groups

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My primary area of research is group theory, and my favorite themes are Coxeter groups, reflections groups and their related structures. My preferred questions take place at the interface of algebraic and geometric combinatorics.******Coxeter groups play a fundamental role in several areas of mathematics: they occur as Weyl groups in Lie theory, for Cluster algebras or in algebraic geometry; they are the discrete reflection groups acting on spaces of constant curvature in geometry and they are fundamental to define buildings in geometric group theory; they also occur naturally in Theoretical Physics, Chemistry and Bioinformatics. Properties of these groups are often key to a deep understanding of the main relevant structures for these areas. ******While there is a vast and rich literature on finite Coxeter groups, or on the role they play, the case of infinite Coxeter groups is, in comparison, still largely unexplored due to lack of explicit combinatorial tools to do so. My research program is about feeling the gap in our understanding of the fine structure of infinite Coxeter groups. ******Together with my collaborators, postdocs and students, I have been at the forefront of a new approach to the combinatorics of infinite Coxeter groups for the past 6 years. Our work provides tools that clarify and reveal profound ties between combinatorial, geometrical and topological aspects of these groups. My current research program, which is a direct continuation of this work, is articulated around the following three axes:******1) Infinite root systems and limit roots;***2) Weak order, Bruhat order and biclosed sets of roots;***3) Garside shadows in Coxeter groups.******At the core of the study of Coxeter groups is a deep connection between their abstract definition and their geometric realizations as reflection groups acting on some geometric spaces. A common technical thread between different components of my research program is the weak order, which has a natural geometric interpretation, and which is as important for Coxeter groups as divisibility is for the integers.******The first topic is about strengthening the bridge that I have contributed to construct between the algebraic combinatorics and the geometric group theory points of view on Coxeter groups******The second topic is concerned with two beautiful conjectures designed to deepen our understanding of Bruhat order, Hecke algebras and Kazhdan-Lusztig polynomials. My motivation in this context is to design a Cluster/Catalan combinatorics theory for the infinite case. ******The last direction is concerned with exploring the notion of Garside shadows that my collaborators and I have made evident in relation to the study of the word and conjugacy problems in Artin-Tits Braid' groups. As consequences, we aim to simplify the description of the automatic structure of Coxeter groups, which aim to open a new perspective on the study of the still open problem of bi-automaticity for Coxeter groups.