Artin groups and CAT(0) spaces

Artin 群和 CAT(0) 空间

• 批准号: 1106726
• 负责人: Ruth Charney
• 金额: $ 28.74万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106726&HistoricalAwards=false
• 关键词: Artin; groups; CAT; spaces

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.

几何群论探讨群与几何之间的相互作用。 中心的兴趣是几何满足一定的曲率条件和集团的作用。这个项目涉及直角阿廷群，一类群之间的自由群和自由阿贝尔群插值。 近年来，这些群在几何群论和低维拓扑学中起着越来越重要的作用。 直角Artin群对满足非正曲率条件（称为CAT（0）条件）的三次复形的作用，对于我们理解这些群以及它们的许多应用都是至关重要的。 此外，这些动作作为理解更一般的CAT（0）空间的丰富的例子。 本项目旨在加深我们对直角Artin群、它们的自同构以及它们在立方复形上的作用的理解，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，为几何对象的对称性群提供了新的见解。对称性在整个科学中扮演着重要的角色，特别是在化学，物理学和天文学中。 这个项目的目标是发展一个更深层次的理解对称群的某种类型的几何对象，被称为立体复杂。立方复形出现在许多情况下。 例如，它们可用于机器人系统建模，以便几何形状的数学属性直接转换为机器人的物理属性。 这是一个不断增长的研究领域，具有许多应用潜力。 此外，PI还参与了旨在吸引年轻人，特别是妇女参与数学的各种活动。 这个项目的想法被用来让这些年轻人参与数学的前沿。