Geometry of Free Group Automorphisms: Beyond the Frontier

群是现代数学许多领域中的一个基本概念。该项目将继续研究自由群的自同构群的动力学和几何学，这是现代数学中的一个关键对象，展示了丰富多样的新的和未探索的现象。该项目将利用该领域最近取得的进展。它还旨在开辟新的研究方向，例如系统地研究自由群的自同态。PI将继续为下一代STEM研究生和本科生做准备，通过举办创新的混合研讨会“几何，群体和动力学”，监督伊利诺伊州几何实验室的垂直整合本科生研究项目，指导数学博士论文研究等。PI还将继续积极参与维基百科的数学主题编辑，旨在接触广大公众和其他科学和学科，并教育他们了解数学研究的现状。该项目将继续发展一个自由群对应的“纤维面”理论的三维流形和研究映射环面的自同构和自同态的自由群。在这项研究中使用的一个关键工具是折叠映射环面相关联的列车轨道图，连同一个半流，然后研究作为一个动力系统，并取代纤维3流形的自由组设置。 具体的问题要调查，包括完善的理解的Bieri-Neumann-Strebel（BNS）不变的映射环面组，构建一个规范版本的McMullen多项式为他们，调查稳定的树和Cannon-Thurston地图，并制定一个系统的火车轨道理论的自由群自同态。

Group is a fundamental concept in many areas of modern Mathematics. The project will continue the study of dynamics and geometry of the automorphism group of a free group, a key object in modern mathematics exhibiting a rich variety of new and unexplored phenomena. The project will capitalize on the recent progress in the area. It also aims to open up new directions of research, such as a systematic study of endomorphisms of free groups. The PI will continue his work on preparing the next generation of STEM graduate and undergraduate students, through running an innovative blended seminar "Geometry, Groups and Dynamics", supervising vertically integrated undergraduate research projects in the Illinois Geometry Lab, directing PhD thesis research in mathematics, and so on. The PI will also continue his active participation in Wikipedia editing on mathematical topics, aiming to reach broad segments of general public and of other sciences and disciplines, and to educate them about the current state of mathematical research. The project will continue developing a free group counterpart of the "fibered face" theory for 3-manifolds and the study of mapping tori of automorphisms and endomorphisms of free groups. A key tool used in this study is the folded mapping torus associated to a train track map, together with a semi-flow, which is then studied as a dynamical system and replaces a fibered 3-manifold in the free group setting. Specific questions to be investigated include refining the understanding of the Bieri-Neumann-Strebel (BNS) invariant for the mapping tori groups, constructing a canonical version of the McMullen polynomial for them, investigating stable trees and Cannon-Thurston maps, and developing a systematic train track theory for free group endomorphisms.