LEAPS-MPS: Functional Identities, Nilpotent Rings, and Garside Shadows

LEAPS-MPS：功能恒等式、幂零环和 Garside Shadows

• 批准号: 2316995
• 负责人: Jordan Bounds
• 金额: $ 19.72万
• 依托单位: Furman University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316995&HistoricalAwards=false
• 关键词: LEAPS; MPS; Functional; Identities; Nilpotent

The study of algebras and Lie theory is a fundamental branch of mathematics that centers around the exploration of symmetries and touches on many aspects of STEM research, including geometry, topology, differential equations, complex analysis, group and ring theory, number theory, and numerous aspects of physics. This PI will focus on nilpotent algebras and a family of abstract algebraic structures known as Coxeter groups. The PI will also integrate the research program with research and training opportunities for undergraduate students with an emphasis on increasing the participation of students from traditionally underrepresented groups in the mathematical sciences. The project will apply functional identity theory to the study of nilpotent rings and algebras, an approach that was successful in the case of general algebras and Lie theory and led to solving Herstein's long-standing open questions concerning Lie and Jordan structures of associative rings. The PI will examine the relationship between nilpotent algebras and Coxeter groups by investigating a relatively new structure called Garside shadows. While it is known that every Coxeter group contains a Garside shadow, there is still much left to determine regarding how these structures relate to the properties of the groups and their associated algebras.This project is jointly funded by the Launching Early-Career Academic Pathways (LEAPS) program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

代数和李理论的研究是数学的一个基本分支，以对称性探索为中心，涉及 STEM 研究的许多方面，包括几何、拓扑、微分方程、复分析、群论和环论、数论以及众多数学理论。物理学的各个方面。本 PI 将重点关注幂零代数和称为 Coxeter 群的抽象代数结构族。 PI 还将把研究项目与本科生的研究和培训机会结合起来，重点是增加数学科学领域传统上代表性不足群体的学生的参与。该项目将把泛函恒等理论应用于幂零环和代数的研究，这种方法在一般代数和李理论中取得了成功，并解决了赫斯坦关于结合环的李和乔丹结构的长期悬而未决的问题。 PI 将通过研究称为 Garside 阴影的相对较新的结构来检查幂零代数和 Coxeter 群之间的关系。虽然众所周知，每个 Coxeter 群都包含一个 Garside 影子，但关于这些结构与群及其相关代数的属性之间的关系，仍有很多问题需要确定。该项目由启动早期职业学术路径 ( LEAPS）计划和刺激竞争研究既定计划（EPSCoR）。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。