The Computational Algebra of Representations of Categories

It is an unfortunate truth in modern mathematics that theory often outpaces practice. More specifically, it is frequently the case that broad theoretical advancements in fields such as algebra are often made without immediate concern for how one might compute many of the objects under discussion. In this project, the PI suggests a plan for advancing a computational framework for an abstract field known as the representation theory of combinatorial categories. As part of the project the PI will also provide several concrete applications of this framework to explicitly computing quantities relevant to researchers in combinatorics, algebra, and topology. Graduate students will be trained as part of this project. More specifically, in seminal work Sam and Snowden developed what is known as the theory of Groebner categories. Roughly speaking, these are categories whose representation theory is well behaved enough to permit something akin to a Groebner basis theory. While Sam and Snowden note that because of the similarities to classical Groebner theory these categories should have a robust computational theory, this computational perspective has not yet materialized. In this project we will fully develop the computational theory of Groebner categories and their representations and provide a number of applications to explicit computations that can be completed in, for instance, the algebraic topology of configuration spaces. 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.