Free Resolutions

This project concerns research in Commutative Algebra. A core goal in the subject deals with understanding the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations. Closely related to this is the concept of a free resolution. Constructing such a resolution amounts to repeatedly solving systems of polynomial equations. For many years, minimal free resolutions have been both central objects and fruitful tools in Commutative Algebra. The idea of constructing free resolutions was introduced by Hilbert in a famous paper in 1890. The study of these objects flourished in the second half of the twentieth century and has seen spectacular progress recently. The field is very broad, with strong connections and applications to other mathematical areas. The broader impacts of the project include the writing of an expository paper, organization of professional development workshops for undergraduate students, and organization of mathematical conferences. The main research goal in this project is to make significant progress in understanding the structure of minimal free resolutions and their numerical invariants. In particular, the PI will: work jointly with M. Mastroeni and J. McCullough on Koszul Algebras; continue work on minimal free resolutions of binomial edge ideals; study the asymptotic structure of minimal free resolutions over exterior algebras. 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.