CAREER: Algebraic, Analytic, and Dynamical Properties of Group Actions on 1-Manifolds and Related Spaces

群是物理对象或理论空间对称性的数学抽象。群是数学中的基本对象，也出现在计算机科学和物理学等各种应用中。群的代数概念与一个集合联系在一起的是一个二元运算，如乘法，它满足一系列公理。群作为数学中各种具体或抽象空间的对称性自然出现。这些空间的几何性质与其对称群的代数性质之间存在着复杂的关系。PI将继续他对无限群体景观的研究，这些群体以数学中最自然的空间，圆和实线的对称性出现。PI将组织两个针对研究生的研究研讨会和两个针对本科生的研究体验项目。这些课程的目标应该是培养多样化的学生群体，使他们成为未来数学领域的领导者。这些活动将把计算方法融入到学生对无限群景观的数学探索中。该项目由拓扑学和促进竞争研究的既定计划（EPSCoR）共同资助。PI将研究左有序群的代数结构与它们在1流形和康托空间上的作用的拓扑和动力学性质之间的关系。一个目标是研究有限呈现的、无限的、简单群的类别，并展示新的概念现象。这包括研究一致简单性等概念，以及是否存在一个有限呈现的无限简单群，它通过同胚作用于实线上。最后，PI将研究组合群论中出现的一系列密切相关的开放问题。这包括系统地研究了有限生成的完美群的正规生成，非指示有限生成的左有序群中的非阿贝尔自由子群的存在性，以及非球面二维CW复的子复的基本群。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A group is a mathematical abstraction of symmetries of a physical object or a theoretical space. Groups are fundamental objects in mathematics that also emerge in various applications such as in computer science and physics. The algebraic notion of a group associates to a set a binary operation, like multiplication, which satisfies a list of axioms. Groups emerge naturally as symmetries of various types of concrete or abstract spaces in mathematics. There is an intricate relationship between the geometric properties of these spaces and the algebraic properties of their groups of symmetries. The PI will continue his investigation of the landscape of infinite groups that emerge as symmetries of the most natural spaces in mathematics, the circle and the real line. The PI will organize two research workshops aimed at graduate students, and two research experiences programs for undergraduates. These shall be aimed at training a diverse body of students to become future leaders in mathematics. These activities will incorporate computational methods into the students' mathematical exploration of the landscape of infinite groups.This project is jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR). The PI will investigate the relationship between the algebraic structure of left orderable groups and the topological and dynamical properties of their actions on 1-manifolds and the cantor space. One goal is to investigate the class of finitely presented, infinite, simple groups, and exhibit new conceptual phenomena. This involves investigating notions such as uniform simplicity, and whether there is a finitely presented infinite simple group that acts on the real line by homeomorphisms. Finally, the PI will investigate a family of closely interconnected open problems emerging in combinatorial group theory. This includes a systematic study of normal generation in the class of finitely generated perfect groups, the conjectured existence of non-abelian free subgroups in non-indicable finitely generated left orderable groups, and fundamental groups of subcomplexes of aspherical 2-dimensional CW complexes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.