LEAPS-MPS: Singularities, Rigidity, and Trace Modules

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The focus on this research is in commutative algebra with motivating questions, and applications, from representation theory and algebraic geometry. Commutative algebra concerns itself with the structure of solutions to systems of polynomial equations and so is connected to fundamental problems in the sciences and engineering.

This project focuses on the study of fundamental algebraic objects known as modules and ideals over commutative rings. In particular, a trace module is the homomorphic image of one module in another and is a generalization of the image of the familiar trace map on a matrix. The trace module is an old construction enjoying renewed interest and has applications in commutative algebra and related fields.

This project involves undergraduate students research with the aim of professional development, preparation for STEM careers, and broadening participation within the mathematical sciences.

This award supports the development of the modern theory of trace modules with applications to the representation theory of rational singularities, the annihilation of cohomology, and Ext-rigidity over commutative rings. In particular, this project is driven by questions under three broad categories related by the theory of trace modules over commutative rings.

The first category explores the representation theory of Arf rings and other singularities. This includes making precise the relationship between the various notions of closure that coincide in the Arf setting with applications to the representation theory of higher-dimensional rings. The second goal is the development of the theory of trace modules over commutative rings.

Finally, the project will study the phenomenon of Ext-rigidity over commutative rings and calculating the cohomology annihilator ideals when cohomology does not vanish.

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.