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

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.