Syzygies and Koszul Algebras

This award supports research in commutative algebra – the study of the set of solutions of systems of multi-variate polynomial equations. Specifically, the project involves the study of free resolutions and Koszul algebras. Free resolutions are technical objects that allow us to approximate complicated algebraic objects by simpler ones. They can often be computed using computer algebra systems such as Macaulay2. Koszul algebras have particularly nice free resolutions and arise in a surprising number of contexts, especially in geometry and combinatorics. As part of this project, the PI seeks to classify certain Koszul algebras in several specific areas of interest. More broadly, the PI will supervise the training of graduate students and postdoctoral fellows. The PI will also begin work on a new textbook on commutative algebra with Macaulay2. A free resolution of a module over a commutative ring is an acyclic sequence of free modules whose zero-th homology equals the module. In the graded setting, resolutions are unique up to isomorphism and encode useful information about the module being resolved. Koszul algebras are graded algebras over a field such that the field has a linear free resolution over the algebra. The PI seeks to establish new classes Koszul algebras related to hyperplane arrangements (via Orlik-Solomon algebras), lattices and matroids (specifically Chow rings and graded Moebius algebras), toric rings (specifically matroid base rings, in connection to White’s Conjecture), and binomial edge ideals. Additionally, the PI will study the Eisenbud-Goto Conjecture in the normal setting, where it is still an open question. 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.