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

职业：1-流形和相关空间上群作用的代数、解析和动力学性质

• 批准号: 2240136
• 负责人: Yash Lodha
• 金额: $ 55.3万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2028-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2240136&HistoricalAwards=false
• 关键词: CAREER; Algebraic; Analytic; Dynamical; Properties

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.

组是物理对象或理论空间对称性的数学抽象。小组是数学中的基本对象，在计算机科学和物理学等各种应用中也出现。组的代数概念将二进制操作（如乘法）相关联，它满足公理列表。小组自然地作为数学中各种类型的混凝土或抽象空间的对称性出现。这些空间的几何特性与其对称性组的代数特性之间存在复杂的关系。 PI将继续对无限群体的景观进行调查，这些群体成为数学，圈子和真实线中最自然空间的对称性。 PI将组织两个针对研究生的研究研讨会，并为本科生提供两项研究经验。这些应旨在培训各种各样的学生成为数学领域的未来领导者。这些活动将将计算方法纳入学生对无限群体景观的数学探索。该项目由拓扑和既定计划（EPSCOR）共同资助。 PI将研究左有序基团的代数结构与其对1个manifolds和Cantor空间的作用的拓扑和动力学特性之间的关系。一个目标是研究有限呈现的，无限，简单的群体的类别，并展示新的概念现象。这涉及调查诸如统一简单性之类的概念，以及是否存在有限的无限简单群体，该群体由同构形态学作用在真实的线上。最后，PI将研究一个在组合群体理论中出现的紧密相互联系的开放问题的家庭。这包括对有限生成的完美群体中正常产生的系统研究，在非诱使有限生成的左有订购组中的非阿布尔自由亚组存在，以及基本的基本群体组的基本群体组的基本群体组成的二维CW复合物，这些奖项反映了NSF的合法传统和范围的依据，该奖项是通过智力和范围的构建范围的，是由智力构建的依据。 标准。