Algebraic and geometric combinatorics of Coxeter groups

我的主要研究领域是群论，我最喜欢的主题是考克斯特组、反思组及其相关结构。我喜欢的问题发生在代数和几何组合的界面上。******Coxeter群在数学的几个领域中扮演着重要的角色：它们在李论、簇代数或代数几何中以Weyl群的形式出现；它们是几何中作用于常曲率空间上的离散反射群，是几何群论中定义建筑物的基础；它们在理论物理、化学和生物信息学中也自然存在。这些基团的性质通常是深入理解这些领域的主要相关结构的关键。******虽然关于有限Coxeter群或它们所扮演的角色有大量丰富的文献，但相比之下，由于缺乏明确的组合工具，无限Coxeter群的情况在很大程度上仍未被探索。我的研究项目是关于感受我们对无限考克斯特群精细结构的理解的差距。******与我的合作者、博士后和学生们一起，在过去的6年里，我一直站在无限Coxeter群组合新方法的最前沿。我们的工作提供了澄清和揭示这些群体的组合、几何和拓扑方面的深刻联系的工具。我目前的研究项目是这项工作的直接延续，它围绕以下三个轴进行阐述：******1)无限根系统和极限根；***2)弱序、Bruhat序和双闭根集；***3) Garside阴影在Coxeter组。****** Coxeter群研究的核心是它们的抽象定义和它们作为反射群作用于某些几何空间的几何实现之间的深刻联系。我的研究项目的不同组成部分之间的一个共同的技术线索是弱序，它有一个自然的几何解释，它对考克斯特群和可整除性对整数一样重要。******第一个主题是关于加强我在代数组合学和几何群论之间所建立的桥梁。******第二个主题是关于两个美丽的猜想，旨在加深我们对Bruhat阶，Hecke代数和Kazhdan-Lusztig多项式的理解。在这种情况下，我的动机是为无限情况设计一个聚类/加泰罗尼亚组合理论。******最后一个方向是探索Garside阴影的概念，这是我和我的合作者在研究Artin-Tits的“辫子”群体中的单词和共轭问题时所明确提出的。因此，本文旨在简化对Coxeter群体自动结构的描述，从而为研究Coxeter群体的双自动性问题开辟一个新的视角。

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.