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

代数和李理论的研究是数学的一个基本分支，围绕对称性的探索为中心，涉及STEM研究的许多方面，包括几何，拓扑，微分方程，复分析，群和环理论，数论和物理学的许多方面。这PI将集中在幂零代数和一个家庭的抽象代数结构称为考克斯特群。PI还将把研究计划与本科生的研究和培训机会结合起来，重点是增加传统上代表性不足的群体的学生在数学科学中的参与。该项目将应用函数恒等式理论研究幂零环和代数，这种方法在一般代数和李理论的情况下是成功的，并导致解决Herstein长期存在的关于结合环的李和约旦结构的开放问题。PI将通过研究一种称为Garside shadows的相对较新的结构来研究幂零代数和Coxeter群之间的关系。虽然已知每个Coxeter群都包含Garside阴影，关于这些结构与群及其相关代数的性质之间的关系，还有许多问题有待解决。该项目由启动早期职业学术途径（LEAPS）计划和刺激竞争研究的既定计划（EPSCoR）联合资助该奖项反映了NSF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。

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.