Geometric Group Theory and Rewriting Systems

摘要Hermiller的proposalGeometric群论和重写系统这个项目涉及三个主要领域的探索领域的几何群论，使用计算技术（特别是，重写系统）和几何方法来研究群体。 第一个领域是几何群论技术在群、幺半群和代数中的扩展和应用。 第二个主题是研究polyfree群，包括这些群的算法问题以及与辫子群和Artin群的连接。 第三个领域是研究良基序和可容许序，包括应用于非交换Groebner基的重写系统的构造。群是一种有用的数学工具，起源于对称性的研究，包括自然发生的物体如晶体和分子的对称性。 例如，将一个正方形映射回自身的平面的反射和旋转形成一个组。 这个项目是在一个领域是在群论，几何和计算机科学之间的接口。 主要研究者的工作集中在这些领域之间的相互作用，特别是追求计算上有效和高效的算法，用于处理与某些几何结构相关的群体。 这个项目还包括这些技术在代数其他领域的应用。

Abstract for Hermiller's proposalGeometric group theory and rewriting systemsThis project involves three main areas of exploration in the field of geometric group theory, using both computational techniques (in particular, rewriting systems) and geometric methods to study groups. The first area is the extension and application of geometric group theory techniques to groups, monoids, and algebras. The second topic is a study of polyfree groups, including algorithmic problems for these groups and connections to braid groups and Artin groups. The third area is a study of well-founded and admissible orderings, including constructions of rewriting systems which have applications to noncommutative Groebner bases.Groups are a useful mathematical tool which originated in the study of symmetry, including symmetries of naturally occurring objects such as crystals and molecules. For example, the reflections and rotations of the plane that map a square back to itself form a group. This project is in a field which is at the interface between group theory, geometry, and computer science. The principal investigator's work focuses on the interplay between these areas, and in particular on the pursuit of computationally effective and efficient algorithms for working with groups that are associated with certain geometric structures. This project also includes applications of these techniques to other areas of algebra.