Artin groups and CAT(0) spaces

几何群论探讨群与几何之间的相互作用。中心兴趣是满足某些曲率条件的几何形状和作用于它们的群。这个项目涉及直角Artin群，一类在自由群和自由阿贝尔群之间插入的群。近年来，这些群在几何群论和低维拓扑学中发挥着越来越重要的作用。直角Artin群在满足非正曲率条件的立方配合物上的作用，被称为CAT(0)条件，是我们理解这些群以及它们的许多应用的核心。此外，这些动作可以作为理解更一般的CAT(0)空间的丰富示例。该项目旨在进一步理解直角Artin群、它们的自同构以及它们对立方复合体的作用，并探索对更一般的CAT(0)空间的影响。几何群论是一个年轻而令人兴奋的领域，它结合了几个数学领域的技术，为几何对象的对称群提供了新的见解。对称在科学中扮演着重要的角色，特别是在化学、物理和天文学中。这个项目的目标是对某种几何物体的对称群有更深的理解，这种物体被称为立方复合体。立方体复合体出现在许多情况下。例如，它们可以用来模拟机器人系统，这样几何的数学特性就可以直接转化为机器人的物理特性。这是一个不断发展的研究领域，具有许多应用潜力。此外，PI还参与了各种旨在使年轻人，特别是妇女参与数学的活动。这个项目的想法正被用来让这些年轻人参与到数学的前沿。

Geometric group theory explores the interaction between groups and geometry. Of central interest are geometries satisfying certain curvature conditions and the groups that act on them. This project concerns right-angled Artin groups, a class of groups that interpolates between free groups and free abelian groups. These groups have played an increasingly important role in geometric group theory and low dimensional topology in recent years. The action of right-angled Artin groups on cubical complexes satisfying a non- positive curvature condition, known as the CAT(0) condition, has been central to our understanding of these groups as well as to many of their applications. Moreover, these actions serve as a rich class of examples for understanding more general CAT(0) spaces. This project is designed to further our understanding of right-angled Artin groups, their automorphisms, and their action on cubical complexes, and to explore implications for more general CAT(0) spaces.Geometric group theory is a young and exciting field that combines techniques from several areas of mathematics to give new insight into groups of symmetries of geometric objects. Symmetries play an important role throughout the sciences, particularly in chemistry, physics, and astronomy. The goal of this project is to develop a deeper understanding of symmetry groups of a certain type of geometric object, known as a cubical complex. Cubical complexes arise in many contexts. For example, they can be used to model robotic systems, so that mathematical properties of the geometry translate directly into physical properties of the robot. This is a growing area of research with potential for many applications. In addition, the PI is involved in a variety of activities designed to engage young people, particularly women, in mathematics. Ideas from this project are being used to involve these young people in the frontier of mathematics.